What is the US proposing to do?
The US says its missile defence system is intended to destroy incoming ballistic missiles potentially coming from North Korea and Iran.
This involves using radars in Alaska and California in the US and at Fylingdales in the UK. Another radar is planned for Greenland.
Anti-missile missiles, or interceptors, are being based in Alaska (40 of them) and California (four).
There would also be 130 interceptors based on ships. The interceptors work by physically hitting the ballistic missile in mid-flight. There would also be missiles to try to destroy incoming rockets in the final stages.
However, the US plans to install 10 more interceptors in silos in Poland, and build a radar station in the Czech Republic.
It hopes that construction of the Czech facility - using a radar currently located at Kwajalein Atoll in the Marshall Islands - could begin next year, with the first interceptors in place in Poland by 2011 and the system fully operational by 2012.
Why in Eastern Europe?
The US says there is a gap in its anti-missile defences.
A threat from North Korea could be countered with the US and sea-based systems.
But European allies or US forces in Europe could be threatened by Iran one day, Washington says, or indeed some other country, so there needs to be a system based in Europe as well.
Will it work?
The theory is that the interceptor missile homes in on and destroys its target in the air by physically hitting the incoming warhead.
However, the closing speed of interceptor and target will be 24,000kph (14,900mph), or 6.5km (4 miles) per second - so the task is more difficult than hitting a bullet with another bullet.
The system's supporters say that not only does it work, but it is even more accurate than that.
But critics say that, despite having spent over $100bn (£54bn), the Pentagon still has not proved the system can work in realistic conditions.
Independent scientific bodies in the US have said that tests of the system's intercept capabilities have been highly scripted, with the defence being given detailed information about the attack beforehand.
They also argue that the defence system could be easily circumvented by potential attackers.
Why does Russia object?
Moscow says that the missiles in Poland and the radar in the Czech Republic could threaten its own defences. The system might be small to start with, it says, but could expand. The radar could be used to spy on Russia.
Russian President Dmitry Medvedev warned that "we will not be hysterical about this, but we will think of retaliatory steps".
The Pentagon says this is just "bellicose rhetoric" from Russia "designed to make Europeans nervous".
Russian objections to the US missile shield grew more strident after the US signed its preliminary deal with Poland in August 2008, at a time when Moscow and the West were already at loggerheads over Russian troops involvement in fighting in Georgia.
A top Russian general said the Poland move would be punished. Moscow's envoy to Nato was quoted as saying that the timing of the deal revealed that Moscow, rather than Iran, was the target of the missile shield plans.
Has Russia proposed any alternatives?
Moscow suggested that the US could use a Russian-rented radar site in Azerbaijan, which shares a border with Iran.
Former President Putin also offered use of a radar site in southern Russia and proposed working with the US and other European countries on a joint defence system.
But the US showed little interest in either idea.
How serious in the threat to Europe?
The Iranian military says its Shahab-3 missiles have a range of 2,000km (1,240 miles).
This means that they could reach south-eastern Europe, hitting targets in Nato members Greece, Bulgaria or Romania.
Russia says that this limited range means that the US missile defence plan is unnecessary, with Foreign Minister Sergei Lavrov saying that negotiations, not threats, were the best way to deal with concerns about Iranian intentions.
The White House said the latest Iranian tests of its Shahab-3 missile do not change the US position on building a European defence shield.
Have Poland and the Czech Republic agreed to the deployment?
It's not yet a done deal.
US Secretary of State Condoleezza Rice has signed an agreement with the Czech government to build the radar station at Brdy, south-west of Prague, but this has to be ratified by parliament.
The Czech government will need the support of some opposition parties to see the measure through. Meanwhile, there is significant public resistance to the plan, with 100,000 people signing a petition calling for a referendum on the issue. Opinion polls indicate that about two-thirds of Czechs do not want the project to go ahead.
Poland signed a preliminary agreement with the US over siting missiles at a Baltic Sea base in August 2008. In exchange, the US agreed to help upgrade Poland's armed forces.
The deal needs further ratification and the centre-left government of Prime Minister Donald Tusk has been considerably cooler on the issue than the previous government, which lost power in October 2007.
What international agreements cover these moves?
None. The US withdrew from the Anti-Ballistic Missile Treaty in 2001.
This treaty limited US and Soviet anti-missile defences to one site each. The Russians still operate theirs, around Moscow.
The US chose to defend its strategic rockets in North Dakota but this defence has been deactivated.
Part of the Russian unhappiness about the Europe sector of the anti-missile system is that it results from the US withdrawal from the ABM treaty and Russia is worried about where it might go next.
Russia has announced the testing of a new multiple-warhead missile, the RS-24, which it says is designed to overcome missile defences. It is also developing new cruise missiles.
The US says it should not be worried.
Is this the start of a new Cold War?
It is certainly a difficult period where mistrust and antagonism are prevalent.
The hopes that Russia and the United States could be friendly allies have not been realised so far.
Instead there is suspicion and this is likely to continue, though to call it a new Cold War is probably going too far.
In May 2008, President Medvedev took over from Vladimir Putin in the Kremlin, and so far he has continued his predecessor vehement disapproval of the US project.
President George W Bush will leave office in January 2009, and the incoming president could take a different line on the issue.
However, both candidates in November's presidential election - Republican Sen John McCain and Democrat Sen Barack Obama - have spoken of the threat posed by Iran.
What ballistic missiles do the US and Russia have?
They have dramatically reduced their arsenals from the Cold War days but still retain substantial forces of several thousand missiles and nuclear warheads each.
Under the Strategic Offensive Reductions Treaty (SORT) signed by presidents Bush and Putin in 2002, each side has to reduce its deployed warheads to a maximum of 2,200 by 2012.
Russia has its own radar early warning system, short-range interceptor missiles in bases around Moscow and a number of land-based Intercontinental Ballistic Missile launch sites across the country.