By Steven Eke
BBC Russian affairs analyst
A Hamas delegation led by the movement's political head, Khaled Meshaal, is in Moscow to meet Russian Foreign Minister Sergei Lavrov and other senior officials.
Russia's decision to invite Hamas for talks has proved controversial
No meeting is scheduled with President Vladimir Putin, despite earlier warnings from Hamas that it would not go to Moscow without one.
Russia is thus the first major power to hold direct talks with Hamas since it won the Palestinian election in January.
Russia's decision to invite Hamas to Moscow without imposing pre-conditions is highly controversial.
It was strongly criticised in Israel, with many politicians suggesting it actually undermined prospects for peace.
Israeli Foreign Minister Tzipi Livni insisted Hamas remained a terrorist group, despite its electoral success, and branded Russia's invitation as "misplaced".
Another minister described Russia's invitation as "stabbing Israel in the back".
The Israeli government convened a special meeting after Moscow made its announcement, to discuss the growing tensions in its relationship with Russia.
In addition to the Hamas invitation, Russia's continuing co-operation with Iran in its nuclear programme worries many Israelis.
The Russian invitation to Hamas initially also caused consternation in Western nations.
Neo-conservative critics in the US accused Russia of betraying the global front against terrorism, while others asked how Russia would react if Israel invited the Chechen rebel leaders to visit Jerusalem.
However, France later backed the proposal, suggesting it might help bring about what French diplomats called "a return to normality in the pursuit of peace".
In addition, the Austrian Foreign Minister Ursula Plassnik, who currently holds the rotating EU presidency, said that while not explicitly endorsing Russia's actions, she saw no reason to doubt Moscow's commitment to the policies of the Middle East Quartet (Russia, the EU, US, and UN).
Russia acknowledges that it has not imposed pre-conditions similar to those demanded by the US and Europe before direct contacts with Hamas. Namely, that the group halts attacks on Israel and recognises its right to exist.
Russia defends this position by saying that it intends to press Hamas to change its stance, precisely by asking it to reject violence and recognise Israel.
But Russia, unlike the US and the EU, does not officially consider Hamas to be a terrorist organisation. This is a crucial difference of policy.
In a recent interview, Russia's envoy to the Middle East, Alexander Kalugin, said Russia would act patiently, without, as he put it, "banging its fists on the table".
He denied that Russia was acting in any way unilaterally, or undermining the Quartet by inviting Hamas to Moscow. He genuinely believed, he said, that Hamas could be transformed into "a positive force".
Support for the Russian policy has also come from the solid body of academic expertise on the Middle East in Moscow.
One leading academic said Russia's natural role was to use the contacts it inherited from Soviet times to positive ends. Supporters of this approach point to how Russia helped bring about an easing of tensions over, say, North Korea's nuclear programme.
Nonetheless, many observers in Israel and the US believe Russia's actions are more about diplomatic grandstanding, and the desire to create the impression that Russia enjoys much more real influence in the Middle East than its status, much reduced since Soviet days, actually allows.
It remains to be seen whether Russia has any ability whatsoever to persuade Hamas to move towards renouncing violence.
But even those supporting the Russian action acknowledge there is no guarantee of results.
For their part, critics point to how Russia's mediation in the stand-off between the West and Iran, over its nuclear programme, has, so far, delivered few, if any, tangible results.