Eusebia Adriana Cosme y Almanza was born in Santiago de Cuba in 1911, the only child of Leocadia Almanza and German Cosme. Her parents died when she was still a minor and she was befriended by a distinguished Santiago family who eventually took her to Habana. In Habana, Cosme studied music and piano theory at the Escuela Municipal de Musica and elocution and declamation at the Academia de Declamacion of the Conservatorio Municipal. It was at the Conservatorio that Cosme received her first professional encouragement from one of her teachers, Graziella Garbalosa.

Cosme's career as an interpreter of Afro-Antillian verse began in the early 1930's when she was asked to recite for Jose Gonzalez Marin, a Spanish actor and reciter. Marin then sponsored her public debut at the Teatro Payret in Habana. Prior to that she had performed only for friends and at school functions.

From the inception of her career, Cosme's recitals were interpretive performances with background scenery and costumes which she designed herself. Her concerts featured primarily the works of Hispanic poets who wrote , poetry with a black theme. However, she also performed the works of Langston Hughes and Paul Laurence Dunbar. Among her favorite poets were Nicolas Guillen, Felix B. Caignet (Cuba), Luis Pales Matos (Puerto Rico), and Andres Eloy Blanco (Venezuela). They, as well as other poets, often wrote works expressly for her. One of these was the now well-known poem, by Eloy Blanco. Additionally, Cosme did dramatic readings from the Afro-Antillian literary genre. She is especially noted for her performance of Hilda Perera Soto's Cuentos de Apolo.

In 1937 Cosme left Cuba to begin her career as an international concert performer. She appeared in South America, the Caribbean, Europe, and the United States. Wherever she appeared she received critical and popular acclaim and was considered the most successful Cuban diseuse of Afro-Antillian verse.

Upon leaving Cuba, Cosme settled in New York City. During the 1940's she had her own radio program there, on Columbia Broadcasting System's Las Cadenas de las Americas, doing dramatic readings and poetry recitals. It was apparently during this period that Cosme married Frederick Laviera.

In 1955 Cosme began her acting career with a Mexican acting company. She appeared in a few plays including a Mexican production of (The Right to be Born), a drama written by Caignet, the Cuban poet/author. The play was made into a movie in 1966 with Cosme in one of the major roles and was an immediate success. For her interpretation of the role of , Cosme was chosen Best Actress and received the “Onix” award, the Mexican version of the “Oscar,” from the Instituto Cinematografico de la Universidad Ibero-Mexicana. Her movie acting career had begun with her appearance in 1964 in Sidney Lumet's as Mrs. Ortiz.

The favorable public and critical response to , and particularly to the character of “Mama Dolores,” led Caignet to write a screenplay entitled “Mama Dolores” (1970) in which Cosme repeated the role. This was followed by acting parts in four other Mexican movies. During this period Cosme resided in Mexico. She also continued to give occasional concert performances in South American and Europe.

Among Cosme's other talents were music composition and painting. She composed the music and lyrics for the theme song to “Mama Dolores,” as well as songs which were sung by popular singers of the day. Her painting career received its impetus from her husband's death in the late 1950's. An abstract painter, Cosme exhibited her works in several of the annual Washington Square Outdoor Art Exhibitions in New York City, as well as in Mexico.

Eusebia Cosme died in Miami sometime in the late 1970s.