The first "team leader", Richard Moser is a hard-bitten cop who, as the first season begins, is going through a bitter divorce from his wife Gina, who takes all their furniture. Moser is also attempting to quit smoking, due to medical problems with circulation. An ex-truck driver, Moser credits Max Koch with keeping him from a life of crime, at one point telling Koch that "I'd be on the wrong side of the law too, just like him", referring to a young pickpocket he has just chased through central Vienna.

Moser has befriended Rex after Rex's former police trainer, Michael, was shot and killed by an escaping suspect; in order to save the dog from being put down, Moser has "adopted" him without ever completing any of the official paperwork. He famously declares at one point, "My taxes pay for this dog, so why can't I give him a better home?". Moser, in contrast to his successors, is shown to be quite athletic, often performing acrobatic stunts while chasing suspects.

As Moser's personal life improves, his sense of humour returns. This is noticeable in the general lightening in the tone of the show from the initial episodes ("Diagnosis Murder" being a prime example) to ones with more light-hearted banter among the officers. Being a bachelor, Moser flirts with many of the attractive women featured in the storylines. However, with the exception of a brief and work-interrupted relationship with his local vet (of which Rex thoroughly disapproves), he forms no permanent romantic attachment.

In the final scenes of the movie Inspector Rex: Moser's Death, Moser is killed in the line of duty by an escaped 'border-line psychopath' played by famous German actor Ulrich Tukur (Das Leben der Anderen) while Moser is rescuing his lover, Patricia Neuhold (a psychologist who has been helping with the case). The escapee commits suicide just after he kills Moser, and Rex shows almost-human emotion over Moser's body at the hospital.