There was much debate as to whether North Korea has nukes or not, until they've finally demonstrated that yes, they do have them, in the early Oughts. Apparently, there were several false starts, including the very first genuine nuclear test that seemed to "fizzle"note , but now they seem to be on track. This pretty much surprised a lot of people, as the general consensus was that North Korea lacks the resources and technology to create a fully functional nuclear device, and that technology gap was projected to close much later.
Still, it looks like their nukes are representative of the Fifties technology that was used to create them, and thus are very bulky and heavy, severely limiting the delivery options. KPAF lacks heavy bombers that could deliver these unwieldy devices to their targets, and these bombers stay little chance against ROK's air defense anyway. TBMs like Scud could be adapted to carry them, and apparently early Taepodong missiles were just that, but they are limited in distance and could barely hit Japan. That's why they are now mostly used as a bargaining chips, with North threatening to conduct a nuclear test each time its foreign relations deteriorate.
North Korea has two missiles that could potentially be used as ICBMs, the Nodong and improved versions of Taepodong rockets, to use their US Reporting Names, in reference to their first spotting location. However, the reliability of these rockets is much debated (the Unha orbital launcher, which failed spectacularly each time it launched,note is almost universally considered to be a barely-modified Taepodong-2). To be honest, thought, failures are pretty normal during testing of the new rockets, and it's not that the North has had many opportunities to test them, given the international pressure. So the rocket launches ended up in the same "bargaining chip threats" bin as the nuclear tests.
...Or not as the case may be. On the 12th of December 2012, North Korea finally succeeded in putting a satellite in orbit. Their three-stage missile flew south over the Yellow Sea and Okinawa before debris crashed into the sea northeast of the Philippines. Seriously though, either this is the last in a long bout of saber rattling, or a sign that North Korea is developing a ballistic delivery system to strike at the west coast of the United States. Or a reminder that even the most unreliable missiles can work properly sometimes.
Things have gotten even more messy as of 2017, when North Korea rapidly modernized their nuclear program and actually got their hands on working ICBMs and even (allegedly) hydrogen bombs. Whether they've managed to build nuclear warheads small enough to actually fit in the missiles remains unknown, but given the rapid improvement of their nuclear technology that's worryingly possible. The fact that the rockets themselves are visibly knock-offs of Cold War era Ukranian designs the North Koreans never had themselves during the Cold War has not gone unnoticed, but that's neither here nor there. They have been testing their missiles much more often than they usually do since then, and have been firing them over Hokkaido, Japan to the ocean beyond, causing major panic in that country and escalating tensions between them, the US and South Korea. While this is mostly still just saber-rattling, it has still caused major alarm regardless.
However, early 2018 saw both Korean leaders began a peace agreement in a historical moment after decades of war with North Korea announcing that it will begin the dismantlement of its nuclear arsenal. There are indications that outside of Western pressure, domestic factors such recent disasters involving their test sites and nuclear bases may have pushed North Korea to drop its plans, though it remains to be seen if they will remain true to their word.