In order of development the following countries are known to possess nuclear weapons:

• United States
• Britain
• Soviet Union (Russia, Ukraine, Kazakhstan)
• France
• China
• India (probably without delivery systems)1974; tested May 1998
• Israel (collaborating with South Africa)
• South Africa (tested 1979 one or more Israeli devices; claims to have dismantled its own; the previous regime may have sent some to Israel before the handover to Majority rule; some may be in private hands). More information on Israel and South Africa here.
• Pakistan (possibly financed by Saudi Arabia and other oil-rich Muslim countries) believed to have achieved a weapon by 1991 and tested in May 1998.
  

• Iraq (factories now destroyed and under special supervision by the IAEA and the US occupation forces)
• Iran (probably with North Korean and Pakistani help, and probably with former Soviet scientists. Kazakhstan may be supplying enriched Uranium).
• North Korea has announced that it possesses at least one device. (Feb 2005).

• Libya - said to have cancelled program following Gaddafi's raprochement
  

• Brazil
• Argentina

• Japan has large quantities of plutonium from civil reactors
  No-one could stop them if they decided to build weapons.

Under the provisions of the 1968 Non-Proliferation Treaty, those with nuclear weapons are not supposed to pass on the techniques to create them to other countries. Other countries with nuclear reactors are supposed to open them to inspection by the International Atomic Energy Agency, part of the UN. The inspection of Iraqi reactors after the war showed that Iraq had evaded the rules, despite having signed the treaty. India and China did not sign the treaty. The Pakistani and Israeli programs show that the treaty can be evaded. Israel and Pakistan are presumed to have had tacit agreement from the United States government (though this would have been illegal both under the Non-Proliferation Treaty and by United States law).
The treaty itself is weak as any signatory can withdraw after three months notice. North Korea did so in March 1993, presumably because it is developing nuclear weapons illegally. Critics of the International Atomic Energy Agency argue that its regulatory functions ought to be separated from its function of spreading "peaceful" technology. That is, one section is devoted to spreading nuclear technology. In the end weapons technology cannot be separated from peaceful technology, which is the usual initial step in starting a weapons program. So far, states can plead sovereignty to keep the inspectors out. One of the needs of a world authority would be to control weapons technology in a serious way.

Following the break up of the Soviet Union there is now a fear that the numerous (30,000) nuclear devices may escape or be sold clandestinely to governments and even guerrilla groups. The unemployed Soviet weapons designers may also migrate and set up in sympathetic countries. Some were reported to have been signed up by Iran. As they were being paid a salary of less than $100 per month at the collapsing rate of exchange there were great temptations for them to use their saleable skills to escape. By 1994 there was evidence of organized criminals attempting to sell Plutonium. Who to? It's not certain.

Strategic Arms Limitation Treaties between the United States and Russia have agreed to reduce existing nuclear weapons to 8000 between them. However these still represent enough power to destroy large parts of the earth. Moreover, it is reported that research on both sides continues to develop new types of weapons, mostly of a smaller size. The danger of these is that the smaller the weapon the greater the willingness to use them on the grounds that the user may not fear that the results will affect him. There is still doubt about whether the other republics of the former Soviet Union will destroy the weapons on their soil or hand them over to Russia. Ukraine was using the weapons as a bargaining tool in its disputes with Russia. The Central Asian states, such as Kazakhstan, may be tempted to assist such countries as Turkey.
If North Korea is proved to possess nuclear weapons there is a danger that South Korea and Japan would then develop them. Both could do so quickly and easily.
What happened to South Africa's weapons? Were they made in collaboration with Israel? Did western countries use SA for clandestine work on neutron bombs? Have any been stolen by extremists?
In June 1995 the new French President Chirac announced France would resume testing (despite outrage among Pacific nations). The US and Britain threatened to follow, thus the nuclear arms race seems likely to continue, not least in South Asia where a new race was sparked off by India and Pakistan's testing of devices in May 1998.

Former President George W Bush was believed to be planning new weapons.

Current President Barack Obama has signed an agreement with Russia to reduce the numbers still further.

North Korea claimed that it had tested a Hiroshima sized bomb on 8 October 2006. It may have exploded another in April 2009.

2009 - President Obama has announced he would like to negotiate the world's nuclear weapons away as they are of no conceivable use in war.