(written from a Production point of view)
Adam Brett Nimoy (born 9 August 1956; age 65) is the son of Star Trek actor Leonard Nimoy, and husband of Terry Farrell. Nimoy was an assistant to Nicholas Meyer on the set of Star Trek VI: The Undiscovered Country and later directed the Star Trek: The Next Generation episodes "Rascals" and "Timescape".
In a short clip in the original series first season blooper reel, Adam, then ten years old in the makeup of a Vulcan. He comes over and says "Hi" to his dad, who is sitting in the bridge command chair, filming a scene of "What Are Little Girls Made Of?".
His father wanted Nimoy to direct him in "Unification II", but this never materialized.
Nimoy wanted to direct episodes of Star Trek: Deep Space Nine, but this did not come to fruition, although Nimoy did get to observe David Carson directing "Emissary". Nimoy commented that "there is a lot of charm to this new offshoot that is attractive to everybody. The characters, the sets, the stories. I'd love to be there. The point is that Star Trek has become like a home to me. I know these people so well, and they're good people, and they're so talented, from pre- to post-production, all the way down the line, this is a really good group of people. It's a winning team and I feel so privileged to be a part of it". (The Deep Space Log Book: A First Season Companion, p 95)
After his father's death, Nimoy announced a Kickstarter fund for a documentary chronicling his father's relationship with Spock titled For the Love of Spock, which then-current Spock performer Zachary Quinto agreed to narrate.  A trailer debuted at the 2016 Film Festival , with the DVD being subsequently released in December that year.
In 2017, Nimoy directed and produced the Star Trek: Deep Space Nine documentary What We Left Behind.  Nimoy commented: "As I travelled around the world with the Spock doc I was blown away by the intense interest in Star Trek: Deep Space Nine. I had no idea there was such a hard core following for the show. The amazing response to our Indiegogo campaign is more proof of the dedication and love of the Trek fans for an incredible show with a phenomenal ensemble cast". 
Nimoy was at first a lawyer, but began a directing career in the 1990s, his apprenticeship with Nicholas Meyer, actually being the very first known employment as such. Nimoy went on to work on such television series as NYPD Blue, Babylon 5, Sliders, The Practice, Ally McBeal, The Gilmore Girls, and The Outer Limits, where he directed his father in the episode "I, Robot".
Nimoy made an appearance in 2015 as himself in the ninth season episode "The Spock Resonance" of The Big Bang Theory, a comedy sitcom series, heavy on Star Trek references. In the episode, art imitated life, as Adam Nimoy was depicted preparing a documentary chronicling the cultural impact his father had as Spock. An excerpt from the episode was subsequently reciprocally featured at the beginning of the later released For the Love of Spock. His father had previously a voice-over part as Spock in the earlier episode "The Transporter Malfunction".