A cell cycle process comprising the steps by which a cell progresses through the nuclear division phase of a meiotic cell cycle. A meiotic cell cycle is the specialized nuclear and cell division in which a single diploid cell undergoes two nuclear divisions following a single round of DNA replication in order to produce four daughter cells that contain half the number of chromosomes as the diploid cell. Meiotic division occurs during the formation of gametes from diploid organisms and at the beginning of haplophase in those organisms that alternate between diploid and haploid generations.