Born in Montgat, Spain, a Catalan fishing village on the Mediterranean coast near Barcelona, Alvar first began painting at the age of 12. He was accepted to the Escuela Superior de Bellas Artes de San Jorge in Barcelona when he was 17. At age 18, he won the Young Painter's Prize in a competition sponsored by the city of Barcelona and his painting was acquired by the Museum of Modern Art in Barcelona for its permanent collection. In 1957, he had his first solo exhibition in the Galleries Layetana in Barcelona. In 1959, after serving in the Spanish Military, Alvar won a scholarship to the Institut Francais in Paris.
In 1960, Alvar met Juan Fuentes, director of the Galerie Drouant in Paris. After Fuentes sold Alvar's first set of paintings in a week, Alvar signed a contract with the gallery. It was at that time Alvar married his childhood sweetheart, Rosella Berenguer, who came to live in Paris with him. He was invited to join the School of Paris, a group of the top young artists in Spain, organized by the Charpentier Gallery. In Paris in the 1960s, he showed with other prestigious Spanish artists including Miro, Dali, and Picasso. In 1962, the Monede Gallery in New York showed Alvar's work in his first exhibition in the United States. In 1963, Alvar produced his first original lithographs for a one-man show at Galerie Drouant. He became known by art collectors for original lithographs and exhibited regularly throughout the United States, Europe, Canada and Japan. Between 1975 and 1990 his creative efforts were focused predominantly on his lithographic works. After ten years in Paris, he returned to Spain where he has lived and worked out of his studio through the present.
When he turned seventy, Alvar transitioned from lithography and rededicated himself to painting in the form of large-scale paintings encompassing a range of subjects from ethereal interiors to biblical narratives.
In 2001, he was commissioned to design a public mixed media mural of the four seasons for Tiana, a suburb of Barcelona. In 2003, he was commissioned to create a permanent public installation in the Plaza de Mallorquines, across from the train station, in Montgat, Spain, a suburb of Barcelona; he designed an 18-foot sculpture titled "Mediterranea." In 2008, he was commissioned to created a sculpture of Catalan cellist Pablo Casals that stands in the Boulogne Billancourt area of Paris.