Aerith and Bob: In a way: most angels and demons have a proper angelic/demonic name and a common Japanese first name (Azazel Atsushi, Beelzebub Yuuichi, Undine Megumi, Salamander Kimitake, Moloch Yoshinobu, Zeruel Takeru, etc.). The angelic/demonic names function as surnames.
Indeed, while Sakuma does get along with Azazel, Beelzebub and most other demons, they don't particularly like her or care about her well-being, and they're sometimes seen plotting her and Akutabe's demise.
Animation Bump: Episodes 5 and 6 feature looser-than-usual animation direction by Chiaki Nakajima, who in the past has worked on The Tatami Galaxy and the token realistic Art Shift episode of Panty & Stocking with Garterbelt. As a result both episodes are some of the best looking in the show, with 6 having some of the most expressive and lively character movement in the TV series, not to mention some interesting scene composition and use of color.
Baleful Polymorph: If you abuse or damage a grimoire, you get turned into an animal. Okada gets turned into a chubby lizard while Seiya becomes a praying mantis. These transformations mean the transformed quickly lose their memories as a human too.
Sakuma is nearly turned into a cow by Moloch (in the first OVA) when she fails to provide him an adequate sacrifice.
Later she actually turns into a panda when Behemoth decides he doesn't want her sacrifice after all. She only turns back when Behemoth accidentally destroys the sacrifice, a PSP.
Berserk Button: Do not question Beelzebub's... peculiar eating habits.
Beyond the Impossible: Many of the things Akutabe does, when he bothers to. One would be smashing into another dimension with just a punch. This is thoroughly lampshaded by the demons.
Bishōnen: Beelzebub's humanoid form. It even becomes a plot point.
He has two, one with insect type add-ons and one without.
Bitch in Sheep's Clothing: Some of the demons. When first introduced, they can be pleasant or polite. However, once we get to know them, their true nature comes out.
Incubus' humanoid form actually has sheep like wool and ram horns. He first seems nice, pleasant, and eager to please. Episode 12 shows a different side to him. His justification for his actions? He's a demon. (Incubi are demons that rape women while they sleep afterall.)
The Behemoth story, big time. A manga editor at Evening (the magazine Azazel-san is published in) wants to murder Kubo Yasuhisa and the entire plot revolves around Evening. Sakuma even likens a manga with half-assed artwork to Hunter × Hunter.
Brought Down to Normal: Inverted; Akutabe allows Beelzebub to take his normal form in the real world by releasing the barrier that makes them into cuddly plush toys. It lasts for two episodes, and then for the rest of it outside the Demon World he's a toy again.
Butt Monkey: Sakuma and Azazel, the latter even more so.
Call Back: In the first episode of season 2, we see a praying mantis and lizard in seperate containers, two previous Baleful Polymorths from season 1.
Council of Angels: Uriel, Raphael, Gabriel and Michael, four (Arch?)angels who seem to be by god's side at all times in the form of four various types of women (The Stoic, Gainaxing, The Pollyanna and Gonk). They are also in charge of making angels fall, cutting off the angel's wings and destroying their halo, before God banishes them to the human world.
Cosplay: Sakuma's forced into this so that she can get some notes for a test. She later joins for real once she finds out there is a profit to be made.
Cute and Psycho: Beelzebub usually comes off as a pleasant (in his own way), well-mannered person... until he slips into the darker side of his persona, which is foul-mouthed, vulgar and violent.
Cute Little Fangs: Azazel's girlfriend, Kiyoko in her humanoid form. Azazel himself has them as well but they're not very cute.
Dangerous Forbidden Technique: Azazel's "The End of Son"note "son"/"musuko" is a slang for "penis" in Japanese attack which works on pulling in all the 'negative' energy and focusing it on an opponent, removing their sexual drive. Borders on Awesome but Impractical as the first time he uses it in an episode he suffers a Heroic Red Ring Of Death, including Blood from the Mouth, and although it had a big start it doesn't seem to do much. It does take effect much later though, turning the tide in the battle. Repeated use just wears him down more and more.
Dark and Troubled Past: Many members of the otaku circle Sakuma joins have suffered severe traumas due to bullying and other abuse in the past.
Deadly Gas: In his monstrous form Beelzebub's breath is comparable to really, really noxious gases. It shreds/dissolves a normal demon's skin.
Demon Lords and Archdevils: There is a hierarchy in Hell, it seems, with different demonic families having one chosen representative with a special power. This representative is the one who gets summoned by a demon summoner. If one dies due to an angel or other ways, the family chooses a new representative to replace the previous one. Some families seem to be more royal than others, like Beelzebub's, but Lucifer just seems to be the strongest Demon there, rather than a leader. Being royalty seems just a title which doesn't mean anything practical.
Despair Event Horizon: Marukome hit this hard, and rather than pulling himself out of it, gets a demon that embodies despair.
Azazel then trumps it by taking this trope Up to Eleven in Episode 12 of the second series. His mother is sleeping with Incubus, he is considered a joke by humans, his girlfriend and mother are both ashamed of him, with his mother even wanting to disown him, and then he sees Incubus get it on with Kiyoko, the closest thing he has to a girlfriend and someone he loves. Incubus was actually raping her, but Azazel doesn't learn of that, and just assumes she was cheating on him. Seems like an extreme case of Break the Haughty as well.
Disappeared Dad: Azazel has a father, it seems, but we've never seen him. He does mention in the second OVA about whether his father is ill, but apart from that it's just his mother.
Salamander's spit is awfully thick. And white. And sticky.
Sariel's hairstyle makes his head look like a penis.
Strawberry Warrior licking the condensed milk off the strawberry in the mock opening...
Deranged Animation: The series is mostly realistic, as per Production I. G.'s usual art style. Oftentimes the animation goes nuts though, almost to Ren and Stimpy levels. Off Model is used to indicate certain changes in someone's personal (Zeruel's animated face seems to literally collapse into a mirage of colours when he finds out he doesn't have a grimoire, and Akutabe is animated (And drawn with a weird expression in the manga) to just a choppy human shaped thing when he tells Undine he loves her.
Impossible Shadow Puppets: God seems to spend his time playing about with this. He is literally just a shadow, and he makes reasonable shapes (A penis, a barking dog, etc) but can apparently emulate defecating and turn his face into a pair of breasts which jiggle.
Played with when Sakuma summons Beelzebub while he and Lucifer are duking it out at full strength, saving the world of demons from serious damage.
The third time Azazel is summoned, he's having sex. The first time Beelzebub is summoned, he's on the john. Which actually makes for a good case of foreshadowing.
Invisible to Normals: The demons by default. They can make themselves visible at will. However, humans with the potential to be Demon Summoners can see them all the time (Azazel speculates it's because they lack a fundamental human trait.)
No Celebrities Were Harmed: Salamander's entire character is a parody of Yukio Mishima. (Although obviously he doesn't look like Mishima, being a small, red lizard, his fundoshi-and-katana outfit is a Shout-Out to Mishima. His full name is Salamander Kimitake. Kimitake was Mishima's real first name.)
Occult Detective: Akutabe. Sakuma begins by studying under him, albeit reluctantly at first, but is now a fully fledged one.
Only Sane Woman: Sakuma, Akutabe to some extent, being always on the spot about things and noticing the details.
Otaku: Sakuma grudgingly cosplays for some in her university to get her hands on some notes. Of course, due to the nature of this show, they are mocked endlessly from all ends of the Otaku spectrum.
Our Angels Are Different: They can't interfere with human affairs unless inflicted by a Demon, and can't use their angelic powers to fight unless they need to. They have wings, but only use them to ascend to heaven, and are charged with finding grimoires in the human world. If an angel takes a grimoire up to Heaven, the demon dies, permanently. The angels currently on earth appear to have a time limit set on finding grimoires, and up above there seem to be 'angel families' (Zeruel has a mother and two cats, for example) and apartment blocks. If an angel fails in getting a grimoire, see Mortality Ensues.
Pandaing to the Audience: Behemoth looks similar to a huge panda (with a trunk), and Sakuma gradually turns into a panda when Behemoth doesn't accept her sacrifice.
Personality Powers: A few demons have this. Azazel embodies lust and is a huge pervert, Beelzebub uses filthy language, Undine is jealous a lot so can make others lives worse, and the Moloch family appears to always be violent, angry muscleheads, among others. Some summoners also end up getting demons with powers appropriate to them (A perverted exhibitionist gets a demon that can make him invisible, although perhaps a subversion while Marukome, who has lost all hope for happy living, gets Eurynome, who embodies despair. Kotaro gets a demon that can eat memories due to the trauma(?) of his grandfather passing away, while the haughty and proud 'President' in the third OVA gets [[Pride Lucifer]], a similarly minded demon). In both cases, however, there's a few subversions too as Eurynome is more about causing despair than being it, and Sakuma is nothing like Azazel or Beelzebub despite having contracts with them.
Punny Name: Akutabe's and Sakuma's names are puns on "aku" (evil) and "akuma" (demon), respectively. The latter becomes a plot point once.
Rape Is a Special Kind of Evil: Incubus rapes Kyoko to mess with Azazel's mind and drive him to despair. Sakuma, up till now becoming more and more like Akutabe with her wanton torturing, destruction and abuse of demons, suddenly takes on a whole new light and demands Incubus explain why. He just responds with 'I'm a demon' and leaves, Sakuma angry and left to comfort Kiyoko. It turns out it was just a hoax.
Rock Me, Asmodeus!: Lucifer tries to demonstrate his awesome guitar-playing skills to Akutabe, but his cute koala paws make it difficult. (He also wears leather pants and sunglasses, and has the ego of a rock star.)
Ship Tease: Akutabe and Sakuma, especially in episode 9 when Akutabe asks Beelzebub and Azazel to go see what Sakuma's up to when her work habits change.
In the manga Azazel implies that Akutabe just pretends to have a hands-off approach to Sakuma learning to be a demon summoner, and is actually watching her from the shadows. (Then again, it's Azazel who says this so it may or may not be true.)
Shout-Out: One of the segments from the ending of episode 10 mirrors the iconic fall of Queen Beryl.
The murder mystery episodes in season 2 features Shallow Parodies of multiple detective characters, including a manzai duo named Hajime and Shinichi. invoked
In episodes 9 and 10 of the second season, the museum thief "48 Faced Freak" is man who runs around in black gloves, boots, mask, and nothing else (though he does wear a bra for some reason), just like a certain Go Nagai heroine. He has bunny ears on his mask, jumps up into air, and can appear anywhere (thanks to his demon's powers).
Stealth Pun: In universe, Moloch gets a show named after him, and it's noted in the anime how the merchandising has gone through the roof with popularity around his character. Get it? He's become a Cash Cow Franchise.
Because of this he is the least popular demon in the manga among female readers. One of the reasons why the director of the anime gave the role to Kazuya Nakai was to boost Salamander's popularity. It worked.
He's so bad, the first thing he does to Sakuma when she greets him is spit on her face. Later, when the man who summoned him introduces himself to her, Salamander chastises him.
Sudden Humility: After spending most of his introduction being a thorough haughty Jerk Ass, Moloch is forced into obeying Akutabe's laws. To wit: He gives a polite 'Itadakimasu', and eats his own kind (Bull demon forced to eat beef). He then proceeds to cry all the way back to the Demon world.
Tsundere: Kiyoko is this with a mix of Kuu Dere for Azazel, smothering him with kisses and proclaiming him her true love in one instance and then threatening to murder him if he doesn't shape up soon enough in the next (as a bonus, she's voice by Rie Kugimiya). Undine also applies in a way, and thinks Akutabe is this thanks to Koutarou convincing her so.