(covers information from several alternate timelines)
The Olympic-class served the Federation as early as the 2370s. (DS9: "Sacrifice of Angels") In an alternate future, the Olympic-class was utilized primarily for medical emergencies. (TNG: "All Good Things...")
The Olympic-class featured a primary section-engineering section-warp nacelle layout common to most Starfleet vessels. However, unlike most Starfleet vessels, the Olympic-class featured a spherical primary hull, similar in outward appearance to the 22nd century Daedalus-class. Its deflector dish was incorporated into the lower forward quarter of the primary hull and was more of a strip than the more traditional shape. Its impulse engines were located on the upper third of the aft hull. A large shuttlebay was situated on the middle upper dorsal section of the secondary hull. (TNG: "All Good Things...")
In an alternate future timeline, the Olympic-class was equipped with sensors and defensive systems that were very limited, leaving the ship to be no match for the weapons of the Klingon attack cruisers of that era. In addition, it was capable of at least warp 13. (TNG: "All Good Things...")
- USS Pasteur (NCC-58925)
- USS Nobel (NCC-55012)
In an interview with Michael Okuda in Journey's End: The Saga of Star Trek: The Next Generation, it was stated, and shown on his computer-generated dedication plaque, that the USS Pasteur was of the Hope-class, named after the WWII hospital ship USS Hope (AH-7). It was later revealed, in the Star Trek Encyclopedia, that the name "Hope-class" had only existed in an early version of the plaque, and was later renamed as Olympic-class. This designation was taken from Bill George having originally named the Pasteur as the USS Olympic. 
An image of the Olympic-class, drawn by Doug Drexler for use in the Encyclopedia, which first appeared on an okudagram in "All Good Things...", also later appeared on a background bridge monitor in DS9: "Sacrifice of Angels".
Design and studio model
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Roots of the Olympic-class' design were laid down when Bill George, over a multitude of years as he grew up, became fixated by one of the earliest design concepts that Matt Jefferies created for the original Enterprise, which George saw in the reference book The Making of Star Trek (pp. 81-82); the defining characteristic of this form of the original Star Trek vehicle was that it had a massive spherical primary hull, which Jefferies seriously considered at the time as he deemed the sphere the best possible pressure vessel for use in a vacuum, but eventually dispensed of as being "too bulky". (Star Trek: The Magazine Volume 1, Issue 10, p. 28)
While serving as a visual effects art director at Industrial Light & Magic (ILM), Bill George suspected that a similar-looking ship might make a nice addition to the final season of Star Trek: The Next Generation and he contacted Michael Okuda to investigate this possibility. Okuda responded by expressing that TNG's production staffers required a starship that was contemporary to the Enterprise-D while still having a unique design. This led George to submit blueprints he did of a ship that was suitably contemporary and had a spherical primary hull. Explaining his thought processes, George has stated on his design, "When I came up with the design for the Excelsior first seen in Star Trek III: The Search for Spock my thought was, "What if the Enterprise had been designed in Japan?" I was very much into Japanese industrial design at the time and tried to put that style into it. My own analysis is that the original U.S.S. Enterprise was Art Deco, and the Star Trek: The Next Generation Enterprise was Art Nouveau, so then the Excelsior would be post-modern. When I designed and built the Pasteur (constructed at home, like my Star Wars Y-Wing), I used a similar idea. I had seen a sketch in The Making of Star Trek years ago of a design idea for the Enterprise that had a spherical primary hull. I thought that if that version had been built, what would the "next generation" update look like? That was the genesis of that design." (Sci-fi & fantasy modeller, Volume 20, p. 82) Though Okuda was initially noncommittal about whether the craft would be used on the show, he commented that doing so was a possibility.
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Bill George, who was formerly an ILM model maker, thereafter worked on the weekends to build a full-fledged studio model based on his artwork. He constructed the miniature around the same time as ILM began working on Star Trek Generations. (The Making of the Trek Films, UK ed., p. 167) "Bill made the model in his spare time as a labor of love," Visual Effects Producer Dan Curry noted. (Cinefantastique, Vol. 25, No.6/Vol. 26, No. 1, p. 64) The model was constructed out of traditional materials wood and styrene, George's modeling materials of choice, with customized neon tubes for internal lighting.
In the script for "All Good Things...", the Pasteur is described as "a small and sleek vessel with the 24th century equivalent of 'Red Cross' markings." However, the model Bill George had built was used for the ship. "[He] called us up to ask if we would like to use it," stated Dan Curry. (Cinefantastique, Vol. 25, No.6/Vol. 26, No. 1, p. 64) George himself continued, "Producer Peter Lauritson ended up renting it from me." (The Making of the Trek Films, UK ed., p. 167) Normally, the series' policy forbade accepting unsolicited models. However, due to extreme time pressures at the time of production on "All Good Things...", Lauritson got dispensation in this one instance. (Star Trek: The Next Generation Companion, 3rd ed., p. 303)
The Olympic-class model has the distinction of being one of only two studio models that were neither commissioned by the studio nor designed or built by Star Trek's own production team (the other one being the Promellian battle cruiser}. Save for the name change from George's original designation of the model as the USS Olympic (NCC-58925) to its ultimately used one as the USS Pasteur (NCC-58925) – though its original name would be retained as class designation – virtually no other modifications were necessary before shooting the motion control photography of the Olympic-class miniature at Image G. 
Several production staffers were pleased that Bill George's model was selected to represent the Olympic-class. "That was really thrilling to see that used," enthused George, "and I was really proud that it was the spherical primary hull that harkened back to that really early design." (The Making of the Trek Films, UK ed., p. 167) Dan Curry remarked, "Bill's model was perfect for the Pasteur, because the spherical front would hold more beds than a sleeker shape like the Enterprise. It's a very stately, peaceful looking ship which is exactly what we needed." (Cinefantastique, Vol. 25, No. 6/Vol. 26, No. 1, p. 64)
Now a bona-fide screen-used Star Trek production item, the model was returned to George who to date still owns his creation. While the model was still residing at Image G, Mike Okuda made use of the opportunity to take a beauty shot of the model for the Pasteur entry inclusion in his Encyclopedia where the model made its second public appearance in the 1997 second and subsequent 1999 third edition, albeit that only a very small version of the picture was reproduced. Years later, in 2011, it was reproduced in its full glory in and on the cover of Volume 20 of the UK Sci-fi & fantasy modeller magazine on the occasion of an interview conducted with Bill George.