Born in Sydney to Norwegian parents, Melgaard was raised in Oslo and studied at the Norwegian National Academy of Fine Arts, moving to the Netherlands in 1991 to complete his studies at the Rijksakademie, Amsterdam, and the Jan van Eyck Academie, Maastricht. The artist lived and worked in New York for a decade, before returning to Oslo in 2017. He represented Norway at the 54th Venice Biennale (2011) and has also participated in the Lyon Biennale (2000 and 2013) and the Whitney Biennial (2014).

His work was the subject of a mid-career retrospective at the Astrup Fearnley Museum, Oslo (2010), followed by Melgaard + Munch – The End Of It All Has Already Happened at the Munch Museum, Oslo (2015), which placed his work in direct dialogue with that of Edvard Munch. The following year, Rizzoli published the first comprehensive monograph of his career. Melgaard's book, A New Novel (2012), was the first English-language novel to be published by Aschehoug in Norway. His most recent novel De Etterlatte will be released in 2021 by the prestigious Norwegian publisher Oktober, that also works with Karl Ove Knausgård.

In 2019, he created his first virtual reality work My Trip with Acute Art, which was shown at the Julia Stoschek Foundation in Berlin. Together with Steiff, he is creating a toy collection to be showcased at Kunsthall Stavanger in 2022, in collaboration with the musician and artist Chris Korda, which includes a retrospective of her career. A major retrospective at the Munch Museum, Oslo in 2023 will be the first comprehensive survey of Melgaard's career at a public institution in his homeland.