What is the timeline from the start of the arms race to the end?

The nuclear arms race between the US and USSR ran from about 1943 (early Manhattan project) to 1991 (end of USSR).

However every time a new weapon is created countries usually enter an arms race to make sure they have better weapons of that type than any other country. An example of this was the introduction of war chariots around 2500 BC, which triggered a series of arms races that lasted past AD 600.