List of countries in order of number of deployed weapons
From a high of 65,000 weapons in 1985, there were about 40,000 nuclear weapons in the world in 2002.
(Statistics from )
Iraq - Had a nuclear weapon research program during the 1980s and 1990s. In 1996, the UN's Hans Blix reported that Iraq had dismantled or destroyed all of their nuclear capabilities. Exact dates remain disputed. 
Japan - Japan conducted research into nuclear weapons during the Second World War.  While Japan has the technological capabilities to develop nuclear weapons in a short time there is no evidence they are doing so. Japan's constitution forbids it from producing nuclear weapons and the country has been active in promoting non-proliferation treaties. There exists some suspicion that there may exist nuclear weapons located in US bases in Japan. 
Kazakhstan - Had 1,400 nuclear weapons, returned them all to Russia by 1995. 
North Korea - On January 10, 2003 North Korea withdrew from the Nuclear Non-Proliferation Treaty. They have since announced their posession of several weapons.
Saudi Arabia - In 2003 members of the goverment stated due to the worsening relations with the USA, Saudi was being forced to consider the development of nuclear weapons. However so far they have denied that they are making any attempt to produce them.
South Africa - Produced 6 nuclear weapons in the 1980's but dissassembled them in the early 1990's. 
Ukraine - Possessed about 5,000 nuclear weapons when it became independent in 1990. It transferred all of these to Russia by 1996. 
Other national nuclear weapons programs
Countries believed to have or sometimes suspected of having at least one unconfirmed nuclear weapon currently, or at some point in history, or research programs with a realistic chance of producing a nuclear weapon in the near future: