Mr Bush met Mr Hu on his visit to Beijing last November
China's leader Hu Jintao arrives in the US on Tuesday for his first official visit as president.
It follows a visit to Beijing by US President George W Bush last November.
The two men will meet in Washington DC on Thursday.
What is the importance of this visit?
The relationship between the world's established superpower, the United States, and a potential superpower of the future, China, is important on several levels.
President Bush has in the past called China a "strategic rival", and the level of their cooperation or competition has yet to be fully decided and is closely watched in the region and around the world.
There are a number of issues that threaten the stability of relations, including the huge American trade deficit with China, the race for energy supplies and the future of Taiwan.
There are also sensitivities over human rights and China's censorship of the internet, seen by the US a prime vehicle for encouraging reforms. Relations are currently calm but edgy. They could break out into recriminations if the trade issue in particular is not addressed.
President Hu Jintao is stopping off to see Bill Gates, the head of Microsoft, first. Why?
Bill Gates is giving a big dinner for President Hu at his house in the other Washington [the state on the West coast], where Mr Hu will arrive in the US.
The event is designed to celebrate an effort by China to address the issue of its use of pirate software and is perhaps also symbolic of the way that trade is so important.
China has recently announced that one of its big computer manufacturers is to buy $250m (£140.9m) worth of Microsoft Windows operating systems.
It might not solve the piracy problem but it is seen as a gesture towards one of the big problems in US-China trade.
What kind of leader is Hu Jintao?
President Hu appears intent on maintaining China's system of capitalism under communism.
China-watchers wonder how long this can survive in its present form but Mr Hu does not seem in any doubt that it can continue.
He is therefore economically quite liberal, but socially restrictive. He has cracked down on the internet, for example.
He is a great traveller abroad (he is also going to Canada and Mexico on this trip) and wants to project the image of China internationally as a dynamic country ready to take its place in the world but also ready to protect its own interests.
Why is the US trade deficit with China so big?
For two reasons. China has become a huge manufacturer of technological products as well as less complex consumer items. It makes everything from hardware to soft toys. US retailers have rushed to get their goods made there because of the low cost.
The second reason is that the Chinese currency, the yuan, is very cheap. Some US officials say it is undervalued by about 40%. That makes the export of goods made in China very cheap as well. Last year, China exported to the US about $200bn more than it imported.
What does the US want doing about the trade deficit?
It would like China to buy more goods in the US (and China has announced the purchase of 80 Boeing aircraft) but more fundamentally it wants China to revalue the yuan, thereby making it more expensive to buy things there.
China has made some moves in this direction but not, in the US view, enough. A US official said recently that China should become a "responsible stakeholder" in the world trade system.
Are the two countries in a race for oil supplies?
All countries these days have to look ahead to ensure their energy supplies and the US and China, being two of the biggest consumers, are sometimes in competition.
The US blocked a Chinese takeover of a US oil company Unocal, which has exploration rights overseas. Some oil experts think that China is now trying to concentrate on getting oil in places where the US has little influence - Iran and Venezuela for example. This has the potential for conflict.
Do they have any interest in stopping global warming?
The US withdrew from the Kyoto agreement on climate control, while China was never part of it. So neither wants to be constrained by targets.
However, partly under pressure from their own peoples, they have both expressed interest in technologies which reduce reliance on damaging emissions and this might be a path they will help each other along in the future.
Do the US and China get along diplomatically?
At the moment, relations are reasonably stable. China rarely causes major problems for the US in the UN Security Council, but the US would like more help from China over North Korea and its nuclear weapons, Iran and Sudan.
Iran, especially, is sensitive. China and Russia oppose sanctions on Iran over its nuclear programme. China has signed a long-term agreement with Iran to buy oil and gas and does not want to upset the Iranians.
China is also interested in oil from Sudan, another potential target of UN sanctions pressed by the US over Darfur.
The US has to pay more and more attention to China, as Chinese world influence grows.
What about the issue of Taiwan?
Both sides want to keep this calm.
China has the ambition of bringing Taiwan back under its control under its "One China" policy. As long as Taiwan does not go for independence, the Chinese ambition is likely to remain a long-term one and it will have patience.
However, if Taiwan did make such moves, China would probably threaten it with military force, though whether it could invade at the moment is doubtful. It is building up its military power significantly year by year.
The US is committed to selling defensive arms to Taiwan and would regard the use of force against Taiwan as a matter "of grave concern" to the US, though it is not formally committed to defending Taiwan in war.