The museum opened in 2011 to a design by architect Rudy Ricciotti, whose work draws on what he calls "hedonist architecture." Ricciotti's design was directly inspired by Cocteau's personality -- the interplay of light and darkness, the self-mythology fueled by contrasts. The building sits barely 50 meters from the Mediterranean shore, a bold geometric structure in black and white that announces itself as something other than a typical Riviera villa. Ricciotti described his intent: "Black and white no longer serve as colours here...they create an interplay of structural forces calling to mind both the artist's works on paper and the poet's personality, his zones of light and darkness." The Menton city council selected his design from an international competition in 2008, and construction began with a foundation stone laid in December of that year.

The museum's collection exists because of Severin Wunderman, an American businessman whose passion for Cocteau bordered on obsession. Wunderman donated nearly 1,800 works to Menton, forming the core of a collection that spans almost 1,000 graphic works divided into seven thematic sections. But the collection's provenance was not without complications. After Cocteau's death in 1963, Annie Guedras cataloged his output and identified at least three dozen fakes and copies among pieces destined for the Wunderman donation. The museum eventually withdrew the questionable works. In 2025, the Wunderman Foundation sent a formal notice to Menton's city council demanding the return of all donated works, arguing that the conditions of the donation had not been respected.

On the night of 29-30 October 2018, a severe storm struck the Menton coast. Waves reached the museum, flooding the basement where management had stored a number of works. Seawater damaged the art, and the ground floor was partially inundated. The museum closed immediately. What was announced as a temporary shutdown for repairs stretched into years of silence. By 2022, the building appeared abandoned, with no official reopening date ever specified by the municipality. Some works were relocated to the Bastion Museum, a small seaside fort built in 1619 that Cocteau himself had helped establish as a museum in 1966. The Bastion, older and more robust, survived the same storm without damage.

While the main museum remains closed, Cocteau's presence in Menton endures in other forms. The Salle des Mariages -- the wedding hall he decorated in the 1950s -- continues to function as both a municipal facility and a tourist attraction, its walls covered with his distinctive figures and Mediterranean scenes. The Bastion Museum displays a rotating selection from the collection. Cocteau, who worked in poetry, film, theater, painting, and drawing with equal restlessness, would perhaps have appreciated the irony: the grand purpose-built museum bearing his name sits empty and embattled, while the modest spaces he decorated by hand remain alive with visitors. Menton's relationship with Cocteau was always more personal than institutional, rooted in the warmth between an artist and a town that recognized each other.