Desi Arnaz invented the rerun during the pregnancy episodes of the series by re-playing some episodes (and change some of the scenery and lines) from the first season to give Lucille Ball time to rest and start to raise their new born son, Desi Arnaz Jr..
Sometimes, Desi Arnaz's distinctive laugh can be heard on the laugh track, especially when he could not control his humor, or laughed extremely quick, just after a deep breath of inhaling air.
Lucille Ball decided to go ahead with the series after having a dream in which Carole Lombard - the screwball comedy actress that died in a plane crash and who was a close friend of Lucy - recommended she take a shot at the risky idea of entering television, and to get off of radio.
William Frawley, aka "Fred Mertz", had a well known and longtime issue with alcoholism. He was advised in the beginning of the series to stay sober, or be terminated. So, if you look closely, a majority of his scenes display his character having his hands deep in his pockets. This would therefore not show his hands trembling, due to his alcoholic withdrawals.
This was one of the first TV shows to be filmed in Hollywood, at a time when many shows were done live in New York. It pioneered the use of three film cameras simultaneously, and the results were high-quality prints of a classic comedy series preserved for future TV audiences.