President Bush's concentration on North Korea's nuclear programme at the meeting of Asian and Pacific countries in Chile shows that stopping the spread of nuclear weapons will continue to be a high priority for the second Bush administration.
North Korea and Iran represent the most immediate major challenges.
Bush wants more talks on North Korea's nuclear programme
The North Koreans have left the Nuclear Non-Proliferation Treaty which is supposed to prevent nuclear weapons from being developed by nations which do not have them. The belief among experts is that North Korea now has several nuclear devices.
The effort to persuade North Korea to change direction is concentrated on the so-called six party talks involving North Korea, the United States, China, Japan, Russia and South Korea.
So far these talks have not produced any progress.
Dr Gary Samore of the International Institute for Strategic Studies in London, who worked on these issues in the Clinton administration, said: "I think the most likely prognosis for the future is that the stalemate will continue.
"North Korea seems absolutely determined to retain a nuclear hedge and for the time being it feels very well protected against the threat of sanctions or any kind of military action."
What nobody knows is what the United States might do if North Korea does not agree to abandon its nuclear programme.
Dr Samore thinks that the stalemate is in the interest of both the US and North Korea right now.
"North Korea may be quite content to just keep on talking without causing any crisis. And Washington as well doesn't really have any good alternative to the six-party talks.
"Sanctions or military force are really not even plausible or attractive options and the US has its hands full in the Middle East."
On the other hand, there is the statement from one of the lower-level hawks in the Bush administration, John Bolton, Under Secretary for Arms Control and International Security.
He wrote in the Financial Times in September: "We are determined to use every resource at our disposal - using diplomacy regularly, economic pressure when it makes a difference, active law enforcement when appropriate and military force when we must"
The risk of attacking North Korea is that a general war might start on the Korean peninsula which would dwarf the war in Iraq.
The question of Iran's nuclear ambitions is an open one.
It says that it has no intention of making a nuclear bomb but it is still ambiguous about whether it might resume the enrichment of fuel for nuclear power one day.
The same technology used for enriching fuel can be used for making weapons grade material.
Iran denies it intends to build a nuclear weapon
Iran was found to have been developing a secret nuclear fuel enrichment programme but has now agreed to suspend this while talks develop with three European countries, Britain, France and Germany, on trade and other incentives.
The issue here is whether the suspension will develop into a permanent agreement. Otherwise the crisis will resurface.
It is not known whether this was a tactical move by Iran to avoid the UN nuclear agency the International Atomic Energy Agency (IAEA) referring it to the Security Council for possible sanctions.
If Iran resumes enrichment, and we will not know until next year, the US may well call for sanctions and if they do not work then the possibility of an air strike by the US or Israel will have to be considered though it would not be easy.
Not only are Iran's nuclear facilties spread out in different locations but an attack would precipiate a new crisis in the Middle East.
There were successes in the first Bush term with the decision by Libya to abandon its secret efforts to develop a nuclear capability and in the rolling up of the private network organised by the Pakistani nuclear expert A Q Khan.
He had been selling nuclear technology, certainly to Libya and possibly to Iran.
There are two other items on the Bush non-proliferation agenda.
The first is to continue its "Proliferation Security Initiative". This is an attempt by a group of like-minded countries to disrupt the sale and shipment of nuclear components, by interceptions if necessary.
One such interception led to the discovery of the Libyan programme.
The other is the next NPT review conference in 2005. The US wants a major loophole in the treaty closed.
This loophole - its "Achilles heel" in the words of the head of the IAEA Mohammed ElBaradei - allows a member state to develop a fuel enrichment ability for its peaceful nuclear reactors.
However that ability could then be used for weapons manufacture.
The US wants enrichment and reprocessing plants to be limited to those countries which already have them.