Azcuy is a Cuban surname. Notable people with the name include:
Adela Azcuy (1861–1914), Cuban nurse and poet Filiberto Azcuy (born 1972), Cuban wrestler Isaac Azcuy (born 1953), Cuban judoka Liván López Azcuy (born 1982), Cuban wrestler Maikel Alejandro Reyes Azcuy (born 1993), Cuban football player Yanqui Azcuy Díaz (born 1976), Cuban boxer Xiomara Rivero Azcuy (born 1968), Cuban javelin thrower