His father was Gustav I Vasa, King of Sweden, his mother Margareta Lejonhufvud.
In 1562 Johan III married Catherine (Katarina) Jagiello in Vilnius. They had a son named Sigismund III Vasa, born in 1566. Since Catherine Jagiello was the sister of Sigismund II of Poland, when Sigismund died without male heir their son was elected king of Poland as Sigismund III in 1587. On John's death Sigismund also gained the Swedish throne.
See also: Vasa Royal family.