Trills are two races that merge to become a single race. They are referred to as the symbiont, a large worm like creature and the Host, the humanoid carrier.
The most well known of all the Trills in the series is Jadzia Dax, a senior science member of the Deep Space 9 crew. Jadzia is a young twenty something female who if it wasn't for the markings down the side of her face, she would be indistinguishable with Earth females.
Not all hosts are selected to be a host. Hosts needs to be selected appropriately. In Invasive Procedures, a rejected Host called Verad played by John Glover (Lionel Luthor in Smallville), hires Klingon mercenaries to kidnap Jadzia. Verad wants Doctor Julian Bashir to move Dax symbiont into him. Julian reluctantly agrees to do so. The Dax symbiont is eventually returned.
The Symbiont can survive in a number of hosts over the years. When a host dies, the symbiont just moves onto the next host. What we would call the surname would be the Symbiont. Before Dax was in Jadzia, Dax was in a male host called Curzon. A symbiont can be placed in either a male or a female body immaterial of where it can been before.
The Host and Symbiont becomes one body, its personality can change over time. Whilst in the body of Curzon, he had been friends with Commander Benjamin Sisko. Benjamin refers to Jadzia as old man on certain occasions. Memories from one host can and do carry over to the other. What you learn in one host is not lost.
Being a host doesn't prevent a host from being a parent, Jadzia remarks when Benjamin is commenting on parenting that she had been both a father and a mother in the past, not always being great. Jadzia is one of the last people to be offerring advice.
During a hosts lifetime, the memories and spirits of previous hosts of the symbiont are transferred to willing participants. It gives the current host to meet the previous ones. Jadzia is able to undertake the ceremony whilst onboard the station with the help of her senior crew members and Quark and Leeta. Jadzia's ceremony is covered in the episode Facets at the end of series 3.
When Jadzia is killed by Gul Dukat, the symbiont is moved across to a new host, Ezri. Ezri is another young female. When Worf tries to make a move on Ezri believing she would have the same feelings, he is rebuffed.
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