Talks during the three-day visit are likely to include a number of issues that are of mutual concern to Russia and the UN, including the status of Kosovo, reforms within the UN Security Council -- of which Russia is a permanent member -- and peacekeeping missions in Afghanistan, Iraq, and Africa.
Foremost among those is the situation in Kosovo, which earlier this year declared independence from Serbia. The tiny Balkan state now wants full UN membership, but its inclusion in the organization is strongly opposed by Russia, which has not recognized its independence.
At talks with Ban at the start of the visit, President-elect Medvedev raised the issue of Kosovo, suggesting that some countries had allowed their support for Pristina to interfere with UN operating standards.
"We think there is no basis for any talk of a crisis or problems at the United Nations. On the contrary, all unified nations should do everything possible to support and develop this universal platform for coordinating their interests," Medvedev said. "Any attempts to resolve international problems in violation of United Nations resolutions -- in particular on Kosovo or other current problems -- are counterproductive and unfortunately do not contribute to reaching peace and stability."
Moscow repeatedly used its position as a permanent member of the UN Security Council to try to block independence for Kosovo. But the United States and some members of the European Union are among dozens of countries that have welcomed Kosovo's new status and support its entry to the United Nations.
The issue of Kosovo has raised questions in Moscow about the role of the Security Council, which hands immense decision-making -- and decision-blocking -- powers to just a handful of countries, including Russia.
"Naturally, Russia is interested in increasing the powers of the Security Council in order to use its power of veto or the threat of this power of veto to make clear its position," says Yevgeny Volk, director of the Moscow office of the Washington-based Heritage Foundation.
Volk adds that Russia feels it has been sidelined in international matters, with other Security Council members, including the United States and Britain, bypassing the council to make important decisions on, say, Kosovo and the invasion of Iraq.
"The last few years have shown that in practice, Western countries, led by the United States and Great Britain, because they are aware of Russia's position [on international matters], have learned to sidestep the mechanism of the UN Security Council and to take their decisions under the auspices of NATO or the European Union, which was the case with Kosovo," Volk says. "In this way, they have managed to limit the role of the UN Security Council and thus ignore Russia's position on many key issues."
Commenting on Ban's visit, Russian Foreign Ministry spokesman Mikhail Kamynin emphasized that Moscow needs a bigger role in the United Nations' governing bodies. "The growing and strengthening role of Russia in international affairs, which is not denied even by the most obvious skeptics, must be reflected by its larger representation in the divisions of the UN Secretariat, including its administration," he said. "We plan to discuss the issue in detail with the secretary-general."
The "Kommersant" newspaper reported on April 9 that Ban was unlikely to be given a smooth ride by Russian officials. Other contentious issues include Iran's nuclear program, which relies heavily on material support from Russia but which is strongly opposed by Western countries.
Russia has agreed to a watered-down version of sanctions against Iran, if it continues its uranium-enrichment program, which other UN Security Council members see as signs of nuclear weapons production.
In an interview on UN Radio this week, the UN secretary-general praised Russia's contribution to UN peacekeeping activities. Ban said the situation in Sudan's Darfur province was something he wanted to discuss with Moscow. It was suggested that he might ask Russia to provide aviation assistance for the troubled region.
Ban also pledged that he would discuss the Middle East peace process with the Russian leadership.
Ban was expected to address students at Moscow State University later this week, and a meeting with the patriarch of the Russian Orthodox Church was also scheduled.