At the conclusion of the Cold War, the US and Russia substantially reduced the size of their nuclear arsenals and signed substantial arms reduction treaties, START I and START II. However, both states still maintain significantly larger nuclear arsenals than any other state. Due to the collapse of the USSR, Ukraine, Kazakhstan and Belarus became nuclear weapons states in the early 90s. The US and Russia both supported these states transferring these weapons to Russia in exchange for commitments to refrain from threatening them militarily.
The US and Russia are both party to the Nuclear Non-Proliferation Treaty (NPT). Under the NPT, states that tested nuclear weapons before 1967 are legally recognised as nuclear-weapon states, but are required to attempt to take genuine steps towards disarmament.