Nuclear weapons can be abolished in the world, but it will take many years and a lot of cooperation. Nuclear devices are VERY expensive to build. Many countries are feeling the pinch of budget cuts from slow economies. Less nuclear weapons are being produced in the world now, than at any time since World War II. It will take cooperation of the international community, to gradually reduce the numbers of weapons that each country has. It's hard to take your finger off the button when so many other world powers have nukes, but even if the US led by example and disarmed most of our weapons as a precursor, we wouldn't be in much immediate danger. In a small finite world, most of the countries that have nukes know that if they use them, there will be fallout, both political and nuclear, effecting them as well.
There needs to be nuclear weapons throughout the world. If everyone has nukes then nobody will fire for fear of the retaliations. If there were no nukes, it is surely possible that some country will disobey the rules. By the time they could be invaded they could already have the weapons.