How will May tackle the challenge of Russia?
Dealing with Russia is one of the biggest challenges for the new PM, but she may use different tactics to her predecessor. David Cameron's relationship with the Russian President Vladimir Putin was prickly at best, poisonous at worst. The former Prime Minister led the charge for sanctions against Russia for its actions in Ukraine. The relationship soured further over efforts to prosecute the killers of the Kremlin critic and former spy Alexander Litvinenko, murdered in London by radiation poisoning in 2006.
The first signals are that she may use rather different tactics to her predecessor in Number 10. On Tuesday the pair spoke in a telephone call initiated, according to the Kremlin, by London. They agreed to meet face to face at the G20 meeting of world leaders in China next month.
The language from the Downing Street statement afterwards was cautious, but relatively warm: "The Prime Minister noted the importance of the relationship between the UK and Russia, and expressed the hope that, despite differences on certain issues, they could communicate in an open and honest way about the issues that mattered most to them. "The Prime Minister and President agreed that British and Russian citizens faced common threats from terrorism, and that cooperation on aviation security in particular was a vital part of the international counter-terrorism effort.
A statement from the Kremlin also hinted at a changing relationship. It said: "While discussing topical issues in Russian-British relations, both sides expressed dissatisfaction with the current state of cooperation in the political, trade and economic spheres." The leaders agreed to intensify "joint work on a number of fronts" including air transportation safety, it added.
The UK's exit from the European Union will inevitably prompt a re-imagining of its relationships with the rest of the world and, for some observers, it's an opportunity that President Putin will want to take advantage of. Attempting to drive a wedge between European countries may play into his hands as the UK looks for closer trade links with nations outside the EU.
When Mrs May took office last month, Mr Putin said he was ready for "constructive dialogue" with the new British leader. He has also said that the UK leaving the EU "will have consequences for the United Kingdom, for all of Europe and for us, of course".