The founding members of the Young Naturalists Society were Edmond S. Meany, J. O. Young, P. Brooks Randolph, and Charles Denny. They wrote a constitution and bylaws in 1880 and began collecting in earnest. Charles Denny's father, Arthur, was a regent of the Territorial University of Washington and arranged for the club to meet on campus. In 1882, a biology instructor named Orson "Bug" Johnson joined the university and brought 20,000 animal specimens with him. Johnson threw himself into the Young Naturalists, and his collection gave the club the largest natural history holdings in the Pacific Northwest. The students built a permanent clubhouse on the university's downtown campus in 1886, funding construction through donations. By the 1890s, when Edmond Meany returned to the university as a history professor and revitalized the group he had co-founded as a teenager, the collection exceeded 60,000 specimens.

When the University of Washington relocated from downtown Seattle to its present campus in 1895, the collection split. Specimens used for teaching moved to the new campus's Denny Hall; the rest stayed in the old clubhouse downtown. In 1899, the Washington State Legislature designated the university's portion as the Washington State Museum, giving institutional permanence to what had been a student project. The Young Naturalists voted to donate their remaining holdings to the state museum in 1904 and disbanded, their mission accomplished. In 1929, Erna Gunther became the museum's director, a post she held for more than 25 years, shaping its identity as a center for Pacific Northwest ethnography and natural science. The museum was eventually renamed for Judge Thomas Burke, a prominent benefactor, and became the official state museum of Washington.

The Burke's collections span anthropology, biology, and geology in staggering breadth. Its ornithology division holds the world's largest collection of spread bird wings. Its frozen bird tissue collection is the second largest on the planet. The mammalogy collection contains more than 55,000 specimens, ranking it tenth among North American university collections. The Northwest Coast ethnographic holdings are the fifth largest collection of Native American art in the world, numbering 10,000 objects that include early collections assembled by Swan, Eells, and Emmons, alongside contemporary silkscreen prints and sculpture. One of the best-preserved Tyrannosaurus rex skulls in the world anchors the "Fossils Uncovered" gallery. A Baird's beaked whale skeleton, from a specimen that washed ashore on a Washington beach in 2015, greets visitors in the main lobby.

The Burke's new home opened on October 12, 2019, replacing a building that had grown too cramped for 16 million objects. Designed by UW alumnus Tom Kundig of Olson Kundig, the $106 million structure broke from traditional museum architecture by putting the work on display. Twelve visible labs allow visitors to watch scientists preparing fossils, cataloging specimens, and conducting research through floor-to-ceiling windows. Storage areas that would normally be hidden behind walls are visible through glass panels. Outside, a garden of 60 native Washington plant species surrounds the building and an adjacent courtyard. The museum's cafe, Off the Rez, serves Native American cuisine, connecting the Burke's deep engagement with Indigenous cultures of the Pacific Northwest to the daily experience of visiting. The old building was demolished in April 2019 after a careful transfer of millions of objects to the new facility next door.