(covers information from several alternate timelines)
The Daystrom Institute, also known as the Daystrom Research Institute, Daystrom Institute of Technology, and Daystrom Technological Institute, was a Federation establishment that existed as early as the 23rd century. It was considered a department within Starfleet. The Institute was named after Doctor Richard Daystrom, who invented the duotronic computer and attempted unsuccessfully to create the first multitronic computer. The institute employed both Starfleet personnel and civilians who were scientific and research specialists in their fields of study. The movement of Starfleet personnel in and out of this organization was governed by the Starfleet Transfer Regulations. The institute was divided into divisions (for example, the Cybernetics Division). (TOS: "The Ultimate Computer"; TNG: "The Measure Of A Man", "Booby Trap", "Data's Day" okudagrams)
The Institute published two periodicals, the Journal of Daystrom Institute and the Daystrom Institute Journal. On stardate 41039, two physicists working at the Institute, Payne and Miller, investigated the spontaneous transformation of tachyons. Their findings supported the hypothesis that tachyon propagation can be described by hybrid-classical interpretations of sub-quantum mechanics. (TNG: "Redemption II" display graphic)
In 2364, the Institute was developing a tomographic imaging scanner capable of multiphasic resolution. By 2370, this device was standard equipment aboard the USS Enterprise-D. (TNG: "All Good Things...")
The Institute had several colleges including the Daystrom Institute of Technology, which was located on Mars. It was attended by Leah Brahms, who earned a Doctor of Theoretical Physics and wrote the thesis Higher Order Warp Field Propulsion Applications. Upon graduation, as a research assistant for the Institute, she developed protocols for measuring higher order subspace distortions and wrote articles for Scientific Tasmanian. After working at the Utopia Planitia Fleet Yards for several years, she became a Professor of Theoretical Physics at the Institute of Technology. (TNG: "Booby Trap" display graphic)
The Institute also included the Daystrom Institute Archaeological Council. Vash was offered a post at the Institute by Professor Woo after she returned from her journeys with Q in 2369. (DS9: "Q-Less")
Seconds before he killed himself by piloting his shuttlecraft into Epsilon 119, Gideon Seyetik asked Benjamin Sisko to send his obituary to the Daystrom Institute for publication. (DS9: "Second Sight")
The identity of this Institute as a department was derived from the regulations that were seen in the extended version of "The Measure Of A Man".
Information on Payne-Miller's experiment was mentioned in the new text that replaced the original text in the remastered episode "Redemption II".
It was suggested in "All Good Things..." that the Daystrom Institute might be located on Rigel III or at least in the Rigel system, as Leah Brahms, the new director in the alternate future, was living on that planet with her family.
Two letters for Jean-Luc Picard from this institute had letterheads imprinted "Daystrom Institute for Advanced Studies". It is suggested in one of the letters that the Daystrom Institute formed the Committee for Quadcentenial for celebrating the Apollo 11 moon landing in 2369. These letters were prepared for the Picard family album, but did not make it on screen.