She was the daughter of Dr. Jorge Guillermo Borges and Leonor Acevedo Suárez. Leonor was given the name Norah by her older brother, Jorge Luis Borges. Of his sister, Jorge wrote:
In all of our games she was always el caudillo, I the slow, timid, submissive one. She climbed to the top of the roof, traipsed through the trees, and I followed along with more fear than enthusiasm. —Jorge Luis Borges, Norah
As a child, she moved with her family to Switzerland to treat the progressive blindness of her father, lawyer Jorge Guillermo Borges. She studied with the classical sculptor Maurice Sarkisoff at the École des Beaux-Arts of Geneva. In Lugano she studied with Arnaldo Bossi and was close to German expressionists such as Ernst Ludwig Kirchner. With Bossi, Borges learned the art of woodcutting and the aesthetics of expressionism.
In Switzerland, Borges wrote and illustrated her first poetry book, Notas lejanas (1915). After the publication, Borges and her family hoped to return to Argentina, but their stay in Europe was extended 4 years because of the First World War. During this time, Borges saw much of Europe. First, she visited Provence (Borges was deeply impressed by Nîmes, and dedicated some of her later work to her travels there). After traveling through Province, she moved to Spain, where she furthered her studies and participated in the Avant-Garde movement. In Spain, Borges first visited Barcelona, and then, in 1919, moved to Palma, Majorca. In Palma, she studied under Sven Westman and collaborated with her brother on the magazine Baleares.
Next she visited Sevilla, where she became a part of the vanguard of Ultraísmo, published her work in magazines like Grecia, Ultra, Tableros y Reflector, and in 1920 she illustrated the cover of El jardin de centauro (The Garden of the Centaur), a book of poems by Adriano del Valle. After leaving Sevilla, she passed through Granada and then finally came to Madrid, where she studied with the painter Julio Romero de Torres. Here she befriended the poet Juan Ramón Jiménez. She illustrated a number of his books and dedicated a portrait to him in her book Españoles de tres mundos.
In March 1921, Borges returned by boat to Buenos Aires. As a young painter, she aligned herself with the vanguard artists of the Florida group. Her work in Prisma reflects the ultraist (anti-modernist) ideas of the group, but her illustrations for magazines such as Mural, Proa and Martín Fierro, and her illustrations in the first edition of the poetry book Fervor de Buenos Aires by Jorge Luis Borges (1923) reveal the influence of the Cubism that she had begun to assimilate with her French contacts in Spain. In 1923, the French surrealist magazine Manomètre, and, in 1924, Martín Fierro published her paintings. In 1926, she displayed 75 works (oils, woodcarvings, drawings, water-colorings, and tapestries) in the Asociacion Amigos del Arte exhibition. In 1928, she married writer and art critic Guillermo de Torre, student of the Ultraist movement and expert on Avant-Garde art and literature, whom she had met in Spain when she was 19 years old. They had two children.
In the Second World War, she became a vocal supporter of la Junta de la Victoria, an association of anti-fascist feminists in Argentina directed by Cora Ratto de Sadosky and Ana Rosa Schlieper de Martínez Guerrero. Also included in the group were the writer María Rosa Oliver, the photographer Annemarie Heinrich, the psychoanalyst Mimí Langer, the artist Raquel Forner, and the poet Silvina Ocampo.
After the war, Norah spent one month with her mother Leo Acevedo in a women’s prison for uttering cries against the president of Argentina Juan Domingo Perón, deepening the aversion that her brother felt for the Argentinian political party Partido Justicialista and its founder Juan Domingo Perón. After her release, Norah illustrated her brother’s book Cuaderno San Martin, as she had done with his earlier works like Luna de enfrente and Fervor de Buenos Aires, and Las invitadas (1961) and Autobiografia de Irene (1962) by Silvina Ocampo. Norah wrote as an art critic for Anales de Buenos Aires under the pseudonym Manuel Pinedo. She worked as a journalist and painter until her death in 1998, but she gave away much of her work and did not care for regular art exhibitions.
In 1942, a version of Platero y yo by Juan Ramon Jimenez was published with illustrations and vignettes by Norah. She also worked as a graphic artist on other books by Spanish emigrants in Argentina like Ramón Gómez de la Serna, Rafael Alberti and León Felipe and illustrated the works of her brother and other Argentinian writers like Victoria Ocampo, Adolfo Bioy Casares, Norah Lange and Julio Cortázar. She designed the scenery of a play by Federico García Lorca using the techniques of oil, watercolor, engraving, woodcut, and drawings in ink and pencil.
She died in Buenos Aires in 1998, aged 97, and was buried in the family vault in the Cementerio de La Recoleta.