All I know was that the Russians nearly didn't make it to the Moon. Sergei P. Korolyov (1906-1998) nearly died after an operation in early 1966, and after he recovered by late April 1966 faced the daunting task of finishing the development of a rocket and spacecraft that could make it to the Moon and back. It was a very good thing that Korolyov decided to launch what could have been Soyuz 1 as the unmanned Kosmos 148 in March 1967, which had all kinds of problems in orbit and crashed back to Earth at a speed of 110 km/h (68 mph) when the parachute only partly deployed, pretty much destroying the landing capsule. That resulted in a massive redesign of the Soyuz spacecraft, and by October 1968, the very successful manned flight of Soyuz 3 cleared one major hurdle for a Moon mission.
The other problem with the N-1 rocket itself. While the complex-looking first stage worked well, the second stage had all kinds of problems, especially with the troublesome NK-15V engines (the first and second test launches in August and December 1967 failed, both due to turbopump failures). It was after the NK-15V got improved turbopumps that the third and four launches succeeded in April and August 1968, and that made it possible for a manned lunar orbital flight in January 1969, a manned Earth orbit test of the LK lander in April 1969, a manned lunar orbit test of the LK lander in July 1969, finally a successful landing in October 1969.