There are five countries that are legally recognized as nuclear weapons states, according to the terms of the Non Proliferation Treaty (NPT), which has been signed by 191 nations: the U.S., Russia, France, U.K., and China.
Additionally, three other countries that are not signatories of the NPT have acquired nuclear weapons — India, Pakistan, and North Korea — while one country, Israel, has not signed the NPT and is not positively known to have nuclear weapons, although it is believed by most analysts to possess them. (For its part, the Israeli government pursues an official policy of “deliberate ambiguity“, in which it refuses to either confirm or deny rumors that it posses nuclear weapons.)
Under a NATO nuclear weapons sharing policy, the U.S. has provided nuclear weapons to Belgium, Germany, Italy, the Netherlands, and Turkey, with the militaries of these countries having the personnel, expertise, and delivery systems needed to maintain and deploy them; however, these arsenals are still technically American, and none of these countries can activate these nukes without the U.S. authorizing it. (Canada and Greece were once part of this program, but ended their involvement in 1984 and 2001, respectively.)
It is estimated that at least forty countries have the resources, technology, and scientific know-how to develop nuclear weapons, ranging from Mexico to Romania. Over a dozen of these countries – including Argentina, Australia, Brazil, Canada, Egypt, Germany, Sweden, and South Korea – at one point had nuclear weapons programs and/or an official policy to acquire nuclear weapons, but formally ended these efforts and disavowed any further pursuit of such weaponry. Four countries – Belarus, Kazakhstan, South Africa, and Ukraine – once possessed nuclear weapons but willingly gave them up. (South Africa actually developed its own nukes during the Apartheid era, whereas the other three inherited them from a dissolved Soviet Union, though did not have the operational capacity to use them.)
At least one country, Iran, is suspected of having a nuclear weapons program of some kind, although its government has taken a strong religious stance against the utilization of these weapons, and most major intelligence agencies worldwide – including all sixteen American intelligence agencies – do not believe Iran has any real aim to arm itself with nukes (though it certainly has the means). A few outdated and isolated reports suggest Saudi Arabia, the United Arab Emirates, and Oman have had some flirtation with acquiring nukes, largely out of concern about Iran.