There are essentially two arguments that people tend to employ when arguing the case for or against nuclear weapons:
That nuclear weapons actually help to preserve peace because people know how terrible the cost of using them would be and are thus deterred from using them so that they aren't treated in a like manner...
That for any country to possess nuclear weapons is completely insane because that nation has the capacity to inflict massive damage on other nations, which could potentially end human civilisation...
Seeing as it's too late to close Pandora's Box, nuclear weapons are out there, and they'll continue to exist. It's completely ridiculous for anyone to think that those nations who currently possess them might decide to relinquish them of their own accord. That's not going to happen.
And because we can't guarantee that the countries that already possess nuclear weapons will ever given them up, it's only logical that nations that don't have them - or at least some of those nations, will be compelled to acquire them, either because they wish to avoid the scales being tipped in the favour of nuclear-weapons armed states in any potential conflict, or because they themselves wish to one day be seen as a heavyweight on the world stage.
One thing to note is that nuclear weapons have only been used in a single conflict. Two were used on Japan in the closing days of the Second World War and modern historians continue to debate why and what purpose their use served. Some think that the Allies were looking for the quickest means to end the war and that the bombs being dropped meant that Allied troops would not have to engage in a lengthy and bloody invasion of Japan... Others think that the Allies were looking to send a message to the Soviets about what might happen if they were to suddenly decide to go in for a land-grab in East Asia or Europe... Still others think that the Allies were taking the opportunity to test the yield and the effects of the new weapon. The truth is probably a combination of all three.
While the argument that nuclear weapons serve as a deterrent is certainly a valid one - especially if one examines the Cold War history of US-USSR relations... The USSR urged the US to sign a "No First Use" agreement numerous times, but the USA refused. Why? Was it because the Americans really and truly thought that they could win a nuclear war against the Soviets? Or was it because the Americans wanted the Soviets to think that they thought they could win?
There are consequences for breaking laws too, but people still do it all time. They jaywalk, they steal, they assault one another, they kill each other, and they are well aware that those things carry penalties. But some people just don't care. They will drive under the influence, they will rape other human beings, they will do things that bring punishment and worry about the consequences later.
Do guns keep the peace? Well, if people know that you have a gun, they're less likely to try and hurt you, rob you, or kill you. But that doesn't mean that there isn't anybody out there who'd be willing to try. And if hurting you, taking something from you, or removing you from the equation might somehow make things better for them, one or two might be willing to take the risk.
Not to mention that if each and every nation possessed nuclear weapons, foreign policy would be completely different. Nuclear weapons states throw their weight around because they know other countries can't threaten them with nuclear weapons, but they don't often do it to each other. Sometimes things get heated - the US and the USSR have engaged in numerous proxy wars over the decades, and Russia continues to work at cross purposes to the US today... The two came right to the brink in the 1960s, and India and Pakistan, sworn enemies, neighbours and nuclear armed states, have fought several times since the 1940s.
But what if a nuclear armed nation really didn't care about the consequences? What if that country was under attack and defeat was inevitable? If the war ending in the enemy's favour was an imminent outcome, what reason would they have NOT to use nuclear weapons?
Would the DPRK not launch its nuclear weapons if it were being overrun by the ROK and USA? Would they not try and hit troop concentrations, air bases and possibly to bomb civilian areas as payback for attacks on their own bases and cities? We can't know. And because we can't know, it's obviously not a good idea to allow politically unstable countries headed by fanatics to possess weapons that are capable of killing tens of millions of civilians and leveling cities.
As weapons evolve, the measures in place to counter them also evolve. For many years it was a question of weighing the merits of initiating a first strike versus responding to one. It was quickly proven that it was much more advantageous to be the one to initiate the first strike in the hope that it might severely dampen the adversary's capability to respond. But because that was dangerous, the weapons and weapons systems were updated and upgraded to ensure that the advantage of a first strike was removed. Nuclear weapons could be stored in such a way that they were essentially invulnerable to a first strike, so you couldn't take them out - even with a direct hit. And because they couldn't be destroyed, they could be readied for use, so that your enemy could hit you just as hard.
Nuclear weapons aren't "good" because the use of even a small number of them would have devastating effects on humanity. But they're "not terrible" in the sense that without the threat of their use, we very likely would have seen some very terrible conventional fighting - tensions flaring up in Europe between the USSR and NATO could have led to a conflict that would have made the Second World War look like a schoolyard fistfight in comparison. It never happened because both sides possessed nuclear weapons.
So I would say that they're probably not going anywhere, unless some total and complete buffoon actually does decide to use them at one point and succeeds in reminding everyone how horrible they are, thereby convincing the rest of the world that they ought to be destroyed... Unlikely, especially at this stage in our development as a species, but not entirely out of the question...
So the best thing that we can do is learn to live with them, and to do that means regulating them. No one who doesn't already possess them ought to be allowed to acquire them. End of story. They are a disease. You cannot allow a disease to spread.