HAZU was founded under the patronage of the Croatian bishop Josip Juraj Strossmayer under the name Yugoslav Academy of Sciences and Arts (Serbo-Croatian: Jugoslavenska akademija znanosti i umjetnosti, JAZU) since its founder wanted to make it the central scientific and artistic institution of all South Slavs.

Today, its main goals are encouraging and organizing scientific work, applying the achieved results, developing of artistic and cultural activities, carrying about the Croatian cultural heritage and its affirmation in the world, publishing the results of scientific research and artistic creativity and giving suggestions and opinions for the advancement of science and art in areas of particular importance to Croatia.

The institution was founded in Zagreb on 29 April 1861 by the decision of the Croatian Parliament (Sabor) as the Yugoslav Academy of Sciences and Arts.

[4] The bishop and benefactor Josip Juraj Strossmayer, a prominent advocate of higher education during the 19th century Croatian national romanticism, set up a trust fund for this purpose and in 1860 submitted a large donation to the then viceroy (ban) of Croatia Josip Šokčević for the cause of being able to bring together the best minds [...] and find a way in which books in the national languages could be produced in the Slavic South; the Academy should also take under its aegis all the areas of human science[5]After some years of deliberations by the Croatian Parliament and the emperor Franz Joseph, it was finally sanctioned by law in 1866.

The official sponsor was Josip Juraj Strossmayer, while the first chairman of the academy was the distinguished Croatian historian Franjo Rački.

The academy's creation was the logical extension of the University of Zagreb, the institution initially created in 1669 and also renewed by bishop Strossmayer in 1874.

The academy has been criticized to the effect that membership and activities are based on academic cronyism and political favor rather than on scientific and artistic merit.

[25] In 2023 the museum restituted to the heirs of Dane Reichsmann artworks that had been looted, including André Derain's “Still Life With a Bottle” and Maurice de Vlaminck's “Landscape by the Water,” as well as lithographs by Pablo Picasso, Pierre-Auguste Renoir, Paul Cézanne and Pierre Bonnard.