Inspired by the comic Tiny Titans, the-dark-cat/The Black Cat not only portrays Batman as a superhero/crime fighting detective, but also as a single father trying to raise his four sons: Dick, Jason, and Tim (the three most famous Robins), and a baby named Terry. The comic is not only funny and adorable, but also succeeds at poking fun at topical points in comics, such as the Dark Knight, Batman's 70th Anniversary, and Grant Morrison. It also occasionally features other DC superheroes and their respective sidekicks/kids.You can read it here or here.
Breaking the Fourth Wall: Baby Terry is surprisingly eloquent. He spends two strips showing off his Genre Savviness and his in-depth knowledge of Batman's comic book history.
In 'Rivalry', the Flash tells Wally that Superman's ability to reverse the rotation of the earth "only happens in the comics".
Also in 'Rivalry', the second and third acts are preceded by characters summarizing the prior events for the audience.
Bullying a Dragon: In the "Rivalry" storyline, Superman tells Lois how Batman always messed with him as a boy, passing it off as him being "such a kidder." When Lois accuses him of being in denial, Superman answers, "It keeps me from killing him, Lois."
Callback: Tim's Superman doll in "Crime and Punishment" made its first appearance two years earlier as one of Tim's presents in the holiday special "Terry's Favorite Thing".
Thomas: First those kids, now (Bruce)'s brought a dog into the manor. Martha, that boy can't do anything right.
Martha:(without looking up from her book) It's nice to know that death hasn't tainted that jaunty personality of yours, Thomas.
Alfred and Jason aren't weak in this department either.
The Faceless: The ghosts of Bruce's parents, and Iris Allen, never have their faces shown.
Five-Finger Discount: Done in "Don't Mess with Bats." After a bad day which gets Batman snarky, he gets on the nerves of Jesse Quick, Hourman and Dr. Mid-Nite by showing his hands "are quicker than" their eyes: he snatches Jesse's top while she runs at him, he gets Hourman's hourglass while he's making his threat (Hourman suddenly sees it's gone and on Batman, who goes "Tick Tock"), and he gets Dr. Mid-Nite's goggles, rendering him unable to see—and running into a tree.
Freudian Trio: Jason is the shoulder-devil Id (particularly the death instinct), Dick is the golden boy Superego, and Tim is hopelessly swayed between them as the Ego.
Hilariously Abusive Childhood: Bruce is very harsh in training the boys (tying them upside down from ropes order to teach them escape skills... while Tim needs to go to the bathroom), though he does try to be fair in his strictness. In "Rivalry", a montage of photos revealed that Bruce had one as well.
Ignored Epiphany: During the Thanksgiving special, Flash reminds Batman and Green Arrow that they are Not So Different and scolds them for ruining the one day Native Americans and settlers got along. The two hang their heads for a moment...and then Batman punches Green Arrow flat.
Mediation Backfire: In "Rivalry", Batman and Green Arrow's latest argument is interrupted by their baby sons hugging. Superman tries to interject An Aesop about the importance of tolerance and friendship, only for the two to turn on him and then return to their argument.
Men Can't Keep House: Averted. While Batman fails dreadfully at chores in the beginning of "Mr. Bat-Mom" due to having servants all of his life, getting a tip to read instructions and then doing research in the Batcave results in the same obsessive perfection that Batman usually gives to detective work.
Also, Barry performs a one-man house-cleaning service in exchange for cake.
The Nicknamer: Batman's WWE-esque nicknames in 'Take Down' range from groanworthy but acceptable (Terror Terry) to nonsensical (Fearsome Tim) to absolutely dreadful (The Dickminator).
Parental Abandonment: Of course, there's the obvious cases (Batman and all three Robins). Also, Catwoman, who is Terry's mother in this continuity, left him in Batman's care, feeling he would be safer with Bruce.
Parental Favoritism: Batman is most overtly affectionate with his youngest (acknowledged) biological son Terry, is also kinder to the youngest ward Tim, tends to be exasperated with Jason's schemes, and holds Dick to very exacting standards. But the true unfavorite? Damian.
Running Gag: In the holiday drawings, Terry hitting Tim with various objects.
Ship Tease: "Rivalry" has numerous examples, mostly based on official comic couples (Roy flirting with Donna, Jade calling Kyle "cute," etc.)
"Crush" has Dick running across town to catch a glimpse of Barbara Gordon at her high school. Then he runs all the way back to get into the Batmobile on time. When Batman asks why he's so sweaty and out of breath, he then notices Starfire walking a few feet away.
Shout-Out: Lots, not just to the comics, but also to the various Batman and/or DC Comics TV series, movies, etc. There's even been references to Peanuts and The Karate Kid.
In-universe: Baby Terry has a habit of climbing to high places, has a Catman doll, and resembles a kitten due to his baby face and short cowl ears. Because his mother is Catwoman.
So Proud of You: Bruce tells Tim in "Rivalry" when Lois gave Tim a ribbon for third place, "I'm proud of you, Tim."
Bruce tells Clark about Terry in 'Super Birthday,' "I'm so proud of him." (Green Arrow's younger son had stolen Terry's balloon, but Terry wouldn't let him get away with it.)
Spinoff Babies: Has shades of this, considering how many DC characters are aged down to kids, babies, or teenagers.
Inverted with some of the veteran JSA members: Alan Scott, Jay Garrick and Wildcat are old to the point of senility (Jay is hinted to not even have Super Speed anymore).
SUNNY: I'd tell you to lick my balls if I still had them!
(Guard dog licks the glass with a loud SLURP!)
SUNNY: (horrified) Dude, he really did it!
BUTCH: (giving middle claw to a smirking guard dog) You Perv!
Thirty Xanatos Pileup : 'Mr. Bat-Mom' consists of Alfred pretending to be have a cold to have Batman take care of the boys for once, Jason and Dick scheming to get Batman back to normal so they won't have to do chores, and Tim working against Jason and Dick because he likes spending time with Batman.
Toilet Humor: Usually one-off jokes based on baby Terry, but also the entire focus of the oneshot "Better Than You".