Its current home was built in 2006 in the South Boston Seaport District and designed by architects Diller Scofidio + Renfro.

[3] In 1937 the Boston Museum of Modern Art moved to its first self-administered gallery space located at 14 Newbury Street and instated a 25 cent admission charge.

'"[3] This same year the newly renamed ICA exhibits works by Le Corbusier in his first show in a United States museum.

In 1959 the ICA installed artwork on the interior of a Stop & Shop on Memorial Drive in a show titled "Young Talent in New England."

[3] 1960 saw the ICA moving to the Metropolitan Boston Arts Center, located at 1175 Soldiers Field Road, which was designed by the museums founder, Nathaniel Saltonstall.

During the five years the ICA spent at this location the museum exhibited, among other things, a collection of works by artists representing the United States at the Venice Biennale (John Chamberlain, Jim Dine, Jasper Johns, Morris Louis, Kenneth Noland, Claes Oldenburg, Robert Rauschenberg, and Frank Stella) and in 1965 the museum housed an exhibit on video and electronic art called "Art Turned On" to which Marcel Duchamp attended.

[3] In 1966 the museum organized an Andy Warhol exhibition with roughly 40 works including selections from Campbell's Soup Cans and portraits of Jackie Kennedy, Marilyn Monroe, Elvis Presley, and Elizabeth Taylor, as well as the first exhibitions in a museum setting of Warhol's films including Eat, Sleep, and Kiss.

[3] 1968 saw the ICA return to the Metropolitan Boston Arts Center, at 1175 Soldiers Field Road, for two years just to move again in 1970 to the Parkman House at 33 Beacon Street as a temporary home.

During these two years the ICA held an exhibit called "Monumental Sculpture for Public Spaces" where large-scale sculptural works by well-known artists, such as Alexander Calder, Donald Judd, Robert Morris, Louise Nevelson, Claes Oldenburg, and Mark di Suvero, were placed in public spaces across the city.

Later in decade the ICA exhibited works by Allan Sekula in his first museum solo show in 1986, held the New England Premiere of the film True Stories by David Byrne, in 1986, who attended the screening, and in 1989 displayed both the first United States survey for Chris Burden as well as the first dedicated major exhibition of the Situationist International movement.

Exhibits have included the first major museum surveys of works by Tara Donovan in 2008, Damián Ortega and Shepard Fairey, who was arrested on vandalism charges on his way to an ICA event,[6] in 2009, and Mark Bradford in 2010.

[3] Formerly located on Boylston Street in the Back Bay neighborhood, the ICA moved to a new facility in the Seaport District of South Boston.

The building's design, which echoes that of nearby waterfront gantry cranes, has been celebrated by many critics for its openness, represented by its exterior grand staircase, and willingness to embrace the surrounding harbor.

It has also been called a "botched box" by architecture critic Philip Nobel, who criticised it for having poor circulation, a dull façade facing land, and casting into shadow the harborside promenade that Elizabeth Diller once referred to as "Boston's only viable civic space.

The planned opening of the 2020 exhibit by Firelei Báez Archived 2020-04-22 at the Wayback Machine was delayed by the ICA's closure, due to the COVID-19 pandemic.

The ICA's exhibition program has included the Momentum series, focusing on the work of emerging artists; the Sandra and Gerald Fineberg Art Wall, an annual, site-specific commission in the museum lobby; the James and Audrey Foster Prize, a biennial exhibition and award for Boston-area artists; and selections from the permanent collection.