After the Nazis deported De Mesquita and his family in February 1944, M.C. Escher went to his teacher's ransacked house and rescued as much of his work as he could. Some of the prints Escher saved still bear the muddy boot prints of German soldiers. De Mesquita, his wife Elisabeth and their son Jaap all died in the camps.

He was born in Amsterdam in 1868, of Portuguese Sephardic Jewish origin. He trained there and was appointed teacher at the School of Architecture and Decorative Arts in Haarlem in 1902, where he remained until 1926. Escher was his most famous student; it was De Mesquita who convinced the young Escher to abandon architecture for graphic arts, a decision that changed twentieth-century visual culture.

De Mesquita produced over four hundred prints, including woodcuts, wood engravings, etchings and lithographs, plus drawings and textile designs. His animal and bird woodcuts, with their stark black-and-white stylisation influenced by Japanese ukiyo-e, are among his most distinctive work. He was not religiously observant despite his Sephardic community background. His wife Elisabeth was gassed alongside him at Auschwitz; their son Jaap perished at Theresienstadt a month later. He died at Auschwitz, around 11 February 1944, at seventy-five.