Martin Réway was born in Prague to Slovakian parents, and started playing ice hockey for HC Slavia Praha; the same team u017digmund Pálffy once played for. Réway not only draws comparisons to the former NHL star, but lists Pálffy as one of his favourite players.
Réway moved to Slovakia in 2009, where he played two seasons for MHK Dolny Kubin before transferring to MHC Mountfield to play on both the U-18 and U-20 teams. He then crossed the border back to the Czech Republic for the 2011-12 season, playing for Sparta Prague's youth teams, as well as making début on the international scene with a spot on Slovakia's U-18 team.