Šulić was born in Maribor, Slovenia. Šulić's father, Božo, is from Dubrovnik, Croatia, and his mother, Alja, is from Izola, Slovenia. His father is also a cellist.

Luka Šulić performs throughout the world combining solo classical performances and with the 2CELLOS.He won series of top prizes at the prestigious international music competitions including first and special prize at the VII Lutosławski International Cello Competition in Warsaw (2009) , first prize at the European Broadcasting Union “New Talent” Competition (2006) and first prize at the Royal Academy of Music Patron’s Award in Wigmore Hall (2011).

He has given a number of solo and chamber music appearances in Europe, South America and Japan in major venues such as Wigmore Hall, Amsterdam Concertgebouw, Vienna Musikverein and Konzerthaus As a soloist he appeared with orchestras such as Deutsche Radio Philharmonie, Australian Chamber Orchestra, Warsaw Philharmonic, Russian Symphony Orchestra and others.

Šulić began his musical education in Maribor when he was five years old. When he was fifteen, he became one of the youngest students ever to enter the Music Academy in Zagreb in the class of Professor Valter Dešpalj where he graduated aged only 18.