After graduating from the old School of San Alejandro in Havana, Cuba, Liliam Cuenca entered the Taller Experimental de Grafica to start working on intaglio printmaking,setting up a professional career working basically on works on paper,drawing and printmaking

She had solo and group exhibitions nationally and internationally and received an award on printmaking, I Triennial of Printmaking ,Havana. Four years later she had her first solo exhibition at the Pequeno Salon of the Museo Nacional de Bellas Artes.

In December of 1980 Cuenca left Cuba and settled in Caracas where she continued her work at the University Simon Rodriguez.

Shortly thereafter, she entered the Taller de Artistas Graficos Asociados, Taga, participating in several groups shows. She had two personal exhibitions at the Eugenio Mendoza Foundation Gallery and was included in the 1983 Program of the Panamerican Graphic Arts, AGPA.

In 1986 she moved to the United States and the following year she is invited to participate in a mayor group exhibition, Recent Development of Latin American Drawing at the Art Institute of Chicago, Then entered the South Florida Art Center, where she received a grant as a residential artist and started painting for the first time.

In 1990 , she won the Cintas Foundation Fellowship and the Asahi Broadcasting Award at the Osaka Triennial in Japan.

She has had solo exhibitions at the Alonso Art Gallery in Bogota Colombia. The Freedom Tower,Miami Dade School for the Arts, Art and Work Space and Cuban Collection in Miami Florida and group exhibitions including The Museo de America, Madrid, Spain. Scriptions. Museo d’Arte Moderna, Villa Crose, Genoa Italy. Confluences, Musee des Beaux Arts de la Ville du Locle  Switzerland. Capitol Gallery, The Capitol. Tallahassee, Florida. The Metro Dade Cultural Resource Center, Miami Florida.

Her work is in the collection in the Housatonic Museum of Bridgeport, Connecticut. The Contemporary Art and Culture Center of Osaka Foundation of Culture, The Oscar B.Cintas Foundation and the Museum of Contemporary Art, Miami, Florida as well as other institutions.