Peter III of Russia
Empress Elizabeth arranged for Peter to marry his second cousin, Sophia Augusta Frederica (later Catherine the Great), daughter of Christian August, Prince of Anhalt-Zerbst and Johanna Elisabeth of Holstein-Gottorp. The young princess formally converted to Russian Orthodoxy and took the name Ekaterina Alexeievna (i.e., Catherine). They married on 21 August 1745. The marriage was not a happy one but produced one son, the future Emperor Paul, and one daughter, Anna Petrovna (20 December 1757 – 19 March 1759). Catherine later claimed that Paul was not fathered by Peter: that, in fact, they had never consummated the marriage. During the sixteen years of their residence in Oranienbaum, Catherine took numerous lovers, while her husband did the same in the beginning.
Anna of Russia
Ivan V had been the older brother of Peter the Great and co-ruler with him, and by that reckoning, his daughters may be considered to have the prior claim. However, if seen from the perspective that the successor should be the nearest kin of the most recent monarch, then the daughters of Peter the Great were nearer to the throne, because they were the aunts of the recently deceased Tsar Peter II. The dilemma was made greater because the daughters of Peter the Great had been born out of wedlock, and had been legitimized later by him, after he formally married their mother Catherine I, who had previously been a maid in his household. On the other hand, Praskovia Saltykova, the wife of Ivan V, had been a nobleman's daughter and a devoted wife and mother; moreover, she had been a lady greatly respected for her many virtues, not least her chastity. 
Finally, the Russian Supreme Privy Council led by Prince Dmitri Golitzyn selected Anna, the second daughter of Ivan V, to be the new Empress of Russia. She was selected in preference to her elder sister Catherine even though Catherine was at that time resident in Russia whereas Anna was not. There were some reasons for this: Anna was a childless widow and there was no immediate danger of an unknown foreigner wielding power in Russia; she also had some experience of government, because she had been administering her late husband's duchy of Courland for almost two decades. Catherine, on the other hand, was married to the Duke of Mecklenburg-Schwerin. She was now separated from him and living in Russia, which was in itself disgraceful; and whether her husband was present or absent, his existence could raise problems at her very coronation. His intervention in government affairs at some later point could hardly be prevented, especially since Catherine had a daughter by him. In that event, since he was ruling prince of ancient lineage with years of experience, he would not be as amenable to the council's advice as a Russian princess. Also, the fact that Catherine had a daughter already would provide a certainty of succession which the nobles perhaps preferred not to have.