The institute derives its name from Tarndanya,[1] the Kaurna Aboriginal people's name for the Adelaide city centre and parklands area, meaning "place of the red kangaroo".

[4] Since then it has hosted a variety of exhibitions and events, including Adelaide Fringe performances and the national launch of the Indigenous Voice to Parliament's Yes campaign in February 2023.

[11] The 2021/2022 annual report showed a loss of $284,000, due to ageing infrastructure and therefore not being able to host Adelaide Fringe events, COVID-19 restrictions, gaps in the workforce, and other reasons.

[15] He was succeeded by Kirstie Parker on an interim basis until late January 2022, when Nancia Guivarra took on the role as acting CEO.

The state government has invested in repairs and renewal of air-conditioning units, sprinkler heads, internal walls, electrical work, painting, and carpets.

[20] Discussions have taken place over several years about the new Aboriginal Art and Cultures Centre (AACC; aka Tarrkarri Centre for First Nations Cultures) as part of the Lot Fourteen precinct on North Terrace;[21] however, a decision on Tarrkarri was still pending in 2024 as the government sought major philanthropic funding for the new site.

[4] During its closure, the space was used to create Still Stylin' 2020 a seven-minute musical video montage featuring reworked songs by Christine Anu.

[13] On 13 November 2020, as part of NAIDOC Week celebrations, the centre hosted a new music festival called TREATY, featuring established and emerging First Nations musicians from South Australia.