(written from a Production point of view)
The Star Trek Smithsonian Exhibit was a museum display opened at the National Air and Space Museum, running for a year from February 1992 through January 1993. Actually, the exhibition was officially intended to be only a part of the larger, overall "Star Trek and the Sixties" exhibit, as it was organized in conjunction with other displays, aimed at highlighting the interaction of Star Trek with other cultural phenomena; yet these were utterly overshadowed by the Star Trek component. (Star Trek and History, Chapter 6) As part of the display, set pieces and costumes from Star Trek: The Original Series were displayed, including the original helm from the USS Enterprise, a tribble, Captain James T. Kirk's Starfleet uniform.  Original studio documentation involving the production of the Original Series, including the original script for "The City on the Edge of Forever", was also part of the exhibit. 
Shortly after the opening, the exhibit expanded in conjunction with the recent release of Star Trek VI: The Undiscovered Country, and production assets, most notably several studio models, from that production were added to the exhibit. Visitors to the exhibit had the chance to view the film on an omnimax film screen.
Especially noteworthy was the inclusion in the exhibit of the restored, actual studio model of the original starship Enterprise, the refit version from Star Trek: The Motion Picture (on loan from Paramount Pictures), the original model for the SS Botany Bay from "Space Seed" as well as the newly restored Class F shuttlecraft model from "The Galileo Seven", the latter two on loan from then owner Gregory Jein.  These models were not easily accessible to the public as they were suspended from the ceiling. Featured in display cases were the original, newly restored D7 class model, the K't'inga-class model (also on loan from Paramount Pictures) , and the original, restored Aurora model from "The Way to Eden". Originally one of the two original Tholian webspinner model from "The Tholian Web", the model was, like the original Enterprise and D7 models, gifted to the museum in 1973.  For the occasion, the museum had the restoration contractor for their possessions, Ed Miarecki, cast an additional copy of the model to represent the original Tholian webspinner as well, as was confirmed by Gary Kerr. (Source) The actual second model was in effect still in existence, as it turned out to be retained by former Original Series set designer John Jefferies, who sold his possession at auction, nine years later. The exhibit marked only the second occasion (after the "1988 Los Angeles Equicon Science Fiction Convention") that a multitude of production-used Star Trek studio models, aside from the original Enterprise, were displayed to a general audience. For the museum's Tholian and D7 models, it has as of 2015 also remained the only time.
A special gallery was reserved to display for the first and only time, the Original Series garments as designed by William Ware Theiss, the gown worn by Leslie Parrish in "Who Mourns for Adonais?" being the eye-catcher.  Theiss' garments were sold off as his estate in the The William Ware Theiss Estate Auction pursuant the Smithsonian exhibit venue, and therefore halfway through pulled from the below-mentioned extension in New York City.
The exhibit ran several Star Trek documentaries made specifically for the exhibit. These documentaries included numerous interviews with Original Series cast members including more rare discussions with such guest cast members as Gary Lockwood and William Campbell. 
Having been the first large specialized Star Trek exhibit, garnering ample contemporary media coverage at the time, the exhibition turned out to be a runaway success. Former "Star Trek and the Sixties" exhibition Smithsonian Advisory Curator H. Bruce Franklin has recalled in this respect, "When "Star Trek and the Sixties" opened, it turned out to be the most popular exhibition in the history of the Air and Space Museum, which had to issue tickets to control the huge influx of people [note: admittance to the Smithsonian museums is usually free]. After more than a million people attended in Washington, the exhibition traveled to the Hayden Planetarium in New York City's American Museum of Natural History to be seen by another huge audience." (Star Trek and History, Chapter 6) The extended exhibition at the Hayden Planetarium opened in July 1993, also running for a year. 
- "Smithonian Trek Photo-Feature", Peter Hardy, Sci-Fi & Fantasy Models, issue 25, January 1998, pp. 53-55
- "Star Trek Props: The 1992 Smithonian Exhibit", Star Trek: The Magazine Volume 2, Issue 10, February 2002, pp. 95-97