Alexis Cuadrado was born in 1971 in Barcelona, Spain. He graduated in Jazz Performance and Composition from Barcelona's Taller de Músics in 1993, and from 1995 to 1998 studied in Paris with the concert bassist François Rabbath. He was already working professionally in these same years: from 1993 he performed across Spain, Europe and the United States alongside musicians including Kurt Rosenwinkel, Bruce Barth, Ben Monder, Perico Sambeat, Jordi Rossy, Allan Skidmore, Dennis Rowland, Louis Stewart, Sean Levitt, Peter King and John Stetch.

Cuadrado moved to New York in 1999. In 2001 he earned a Master's degree, graduating with honors, in Jazz Performance and Composition from the Aaron Copland School of Music at Queens College, where he also studied with bassist Larry Grenadier. That same year he released his debut as a leader, Metro (Fresh Sound New Talent), recorded in New Jersey with drummer Mark Ferber, guitarist Steve Cardenas and pianist Pete Rende. A second leader date, Visual (2004), billed to the Alexis Cuadrado Sextet, kept Ferber and Cardenas and added saxophonist John Ellis and trombonist Alan Ferber.

In 2008 Cuadrado co-founded Brooklyn Jazz Underground (BJU), a collective and small record label built around a cohort of Brooklyn-based jazz musicians, and released his third leader album, Puzzles, on the label's bjurecords imprint that year, with saxophonist Loren Stillman, guitarist Brad Shepik and drummer Mark Ferber. He also appeared on several of the label's own Brooklyn Jazz Underground compilation volumes.

A 2008 Chamber Music America New Works grant, administered through the Doris Duke Charitable Foundation, funded Cuadrado's fourth album, Noneto Ibérico: a nine-piece suite that sets each movement to a specific flamenco song-form, or palo, reworked into jazz harmony and structure. The piece premiered in October 2009 and the recording was released on bjurecords in 2011. The nonet included trumpeter Avishai Cohen (Trumpet), guitarist Brad Shepik, saxophonist Loren Stillman, trombonist Alan Ferber, pianist Dan Tepfer, percussionist Marc Miralta and three flamenco handclap (palmas) specialists, Blas Córdoba, Israel Suárez "Piraña" and Tomás Moreno. Reviewing the album for All About Jazz, Mark F. Turner wrote that it "reboots flamenco-jazz for the twenty-first century."

A second Chamber Music America commission, through the organization's New Jazz Works program, followed in 2011 and led to A Lorca Soundscape (Sunnyside, 2013), a suite setting poems by Federico García Lorca for a chamber-jazz quintet of voice, alto saxophone, piano, bass and drums. The record featured vocalist Claudia Acuña, alto saxophonist Miguel Zenón, pianist Dan Tepfer and drummer Mark Ferber, and was performed live at New York's 92Y Tribeca.

Cuadrado's sixth album as a bandleader, Poètica (Sunnyside, 2016), turned to English- and Catalan-language poetry, setting texts by Rowan Ricardo Phillips and the Catalan poet Melcion Mateu for a quartet of guitarist Miles Okazaki, pianist Andy Milne and drummer Tyshawn Sorey, with Cuadrado himself also reciting and singing some of the words. Alongside his own recordings, he spent fifteen years, from 2007 to 2022, as an Associate Professor at the New School's College of Performing Arts in New York, and worked widely as a sideman and producer, including with vocalists Dianne Reeves, Angélique Kidjo, Youssou N'Dour and Omara Portuondo, pianist Chano Domínguez, violinist Regina Carter and pianist Jon Batiste. In 2021 he stepped back from performing to found The Plaza Roja, a Brooklyn studio for film and media scoring, and from 2022 to 2024 worked as a senior music producer at the sonic-branding company Made Music Studio.

Cuadrado's music moves between straight-ahead post-bop, the flamenco vocabulary of his native Spain, and a more open, through-composed idiom shaped by his conservatory training as a composer as much as by his work as a bassist. His concert-music leanings are most explicit on Noneto Ibérico, where he set each movement to a distinct flamenco palo and reworked its traditional rhythmic cycle into large-ensemble jazz writing; reviewing the record for All About Jazz, Mark F. Turner praised its "complex yet accessible" charts and likened Cuadrado's compositional sophistication to that of the Argentine composer-bandleader Guillermo Klein. Cuadrado has named Maria Schneider, Darcy James Argue and Klein as the contemporary composers he follows most closely, alongside older influences ranging from Mozart and Beethoven to the minimalism of Philip Glass and Steve Reich, and, by his own account, the genre-hopping sequencing of the Beatles' White Album. His two Chamber Music America commissions, for Noneto Ibérico and A Lorca Soundscape, both grew out of the same instinct to treat a non-jazz text or tradition, Andalusian flamenco song-forms in one case, García Lorca's poetry in the other, as the structural spine of an extended jazz work rather than as decoration on top of one.