Both leaders stressed their new trusting relationship
Speaking after talks with his Russian counterpart Vladimir Putin, President Bush said the US would reduce operational nuclear warheads from about 7,000 to between 1,700 and 2,200 over the next decade.
The two leaders also discussed US plans for missile defence and the subsequent fate of the bilateral Anti-Ballistic Missile (ABM) treaty.
But there was no agreement on this subject and President Putin said the two leaders needed to continue talks at President Bush's ranch at Crawford, Texas on Wednesday and Thursday.
Both men stressed that the relationship between their countries had transformed from that of the Cold War era. President Bush said it had changed "from one based on suspicion to one based on trust".
'Cuts appreciated'
The Russian leader said he appreciated the cuts in the United State's nuclear stockpiles and said Russia would try to respond in kind.
Russia is known to be keen to dismantle a large number of their own missiles because they are so expensive to maintain.
Presidents Bush and Putin have been brought closer together by the 11 September attacks on the US.
President Putin was the first world leader to send his condolences, and the two men told the press they would co-operate to beat terrorism.
Afghanistan
They also found common ground on the issue of a future government in Afghanistan - President Bush said he and President Putin backed the UN call for a broadly-based and multi-ethnic administration in the ravaged country.
However, the BBC's Philippa Thomas noted that while President Bush said that the Northern Alliance would find no preferential treatment at the negotiating table, Russia has been supporting the Alliance for years - a source of potential tension.
President Bush said the talks heralded "a new
day in the long history of Russian-American relations, a day of progress and a day of hope."
Reflecting this new-found trust, the US leader said there was no need for "endless" discussions on arms controls.
"I looked the man in the eye and shook his hand. But if you need to write it down on a piece of paper I'll be glad to do that," he told President Putin.
Mr Putin was not quite so relaxed, stressing that there had to be a "reliable and verifiable agreement" on cutting arms.
Cold War history
President Bush also said he would work to end Cold War-era restrictions on bilateral trade.
More than 40 years of the Cold War, during which the two sides bristled with tens of thousands of nuclear weapons in a dangerous stand-off, bred mutual suspicion and distrust.
The ABM treaty, signed in 1972, was designed to lighten some of this tension, as it prohibited the US and the Soviet Union to manufacture defences against potential missile attacks.
It is this treaty which the US now wants to scrap, arguing that it is outdated, but Russia argues that is remains a cornerstone of global security.