They met today. From AsiaNews:

Relations between Catholics and Orthodox were also on the agenda of talks between Russian President Vladimir Putin and Benedict XVI, which took place this evening in the Vatican, in a “very positive climate”.  The “cordial relations” between the Vatican and Federation as well as international concerns, in particular the current Middle East crises, were other themes broached in discussions, which also touched upon the “problems of intolerance and extremism”, according to a press office statement released by the Holy See.

The Vatican, then has judged the first meeting between Benedict XVI and Putin to as positive, which also saw a parallel encounter between Russian Foreign Minister Lavrov and Secretary of State Tarcisio Bertone. “The meeting – reads the Vatican note – which took place in a very positive atmosphere, highlights existing relations between the Holy See and the Russian Federation, as well as the shared desire to further develop said relations, through specific cultural initiatives.  In light of this, bilateral themes of common interest were discussed with careful consideration also paid to relations between the Catholic Church and Orthodox Church.  Current international questions were also analyzed, in particular the Middle East. Finally, the problems of intolerance and extremism were addressed, which constitute a grave threat to civil relations between Nations, underlining the necessity to preserve and favour peace through negotiated solutions to ongoing conflicts”.

Even Russian journalists in the presidential cortege speak of a “climate of great cordiality, beyond protocol, sign of a significant event”. The Russian Presidential spokesman, Aleksiei Gromov, told the press that Putin had referred greetings from the Patriarch of Moscow Alexi II to the Pope, who likewise extended a salute to the leader of the Russian Orthodox Church.

Putin arrived at the Vatican shortly before 6 pm, dressed in a blue suit and red tie. He was met in the St Damascus courtyard by the Papal gentlemen; he was escorted to the third floor, passing through the halls which lead to the private library and the Pope’s study. Benedict XVI greeted the Russian President moving forward to meet him in the library, in the adjacent Throne room.  “I warmly welcome you to the Vatican”, he said to him in German, a language which Putin knows well, and a handshake which journalist present describe as “long and cordial”. “Now, we must allow for photographs” added the Pope, after Putin positioned himself at the desk, the same place occupied by Gorbachev in his historic visit with John Paul II and then the doors closed on the private meeting which lasted over 25 minutes.  Putin and Benedict XVI spoke in German assisted by two interpreters.

Following the encounter, there was the presentation of the delegations and the exchange of gifts. The Russian President presented the Pope with a modern icon of St Nicolas.  The Pope for his part, gifted the premier with an incision of a view of St Peter dated to 1662.