Fucsovics started to play tennis at the age of five. He came through the ranks quickly and at the age of 8 he played in a field 3–4 years older.
In 2003 Fucsovics won his age group National Championship and the Nike Junior Tour. With that he got the chance to represent Hungary on the World Final in Sun City, South Africa. Fucsovics also showed talent in basketball as a youth, but chose to make tennis his career path.
Fucsovics was also an integral part of the Hungarian U-18 national tennis team. The team in fact won the prestigious Galea/Valerio Cup in Venice to claim Hungary's first ever boys European Summer Cups title. The other members of the team were Máté Zsiga and Levente Gödry.
Fucsovics worked his way to the top of the ITF Junior Rankings and as a result entered the Youth Olympics as tournament's top seed. He suffered a surprise loss, however, in the first round, to eventual quarterfinalist Oliver Golding (7–5, 6–3).
Won tournament; or reached Final; Semifinal; Quarter-final; Round 4, 3, 2, 1; competed at a Round Robin stage; lost in Qualification Round; absent from tournament event; played in a Davis Cup Zonal Group (with its number indication) or Play-off; won a bronze, silver (F or S) or gold medal at the Olympics; a downgraded Masters Series/1000 tournament (Not a Masters Series); or a tournament that was Not Held in a given year.
To avoid confusion and double counting, these charts are updated either at the conclusion of a tournament, or when the player's participation in the tournament has ended.