Cudia is noted for his critically acclaimed portrayals in Andrew Lloyd Webber's The Phantom of the Opera.
Cudia stayed with the touring company until the spring of 2008 when he left to portray Valjean in the Marriott-Lincolnshire Theater's production of Les Misérables.
Cudia played his first performance in the ten-week stint on May 22, 2008, and continued until McGillin returned on July 28, 2008.
Cudia then became the "principal" Phantom at the Majestic Theatre, having taken over from Howard McGillin on July 27, 2009.
He made his Lyric Opera of Chicago debut as Curly in its production of Oklahoma!.