Mystique is one of the better-known villains, anti-heroes, and sometimes heroes in The X-Men.

She founded the Brotherhood of Mutants and has assassinated a number of important people. She was introduced in Ms. Marvel #16 (May 1978)When she was created, writer Chris Claremont wanted to explicitly show that she was in a relationship with Destiny. Marvel would not allow it, as the comic code did not allow the portrayal of same-sex couples at that time.[1]

There are, however, hints that her close friendship with Destiny is more than friendship. In 1988, they are allowed to share a romantic dance with on another in Marvel Fanfare #40.

At one point in her history, Mystique also spends time in Germany married to Baron Christian Wagner. She uses her shapeshifting powers to have various affairs at that time. One of those is with fellow mutant Azazel and she becomes pregnant. Her child is Kurt Wagner, also known as Nightcrawler. 

Mystique holding Irene with a plain background. Mystique is smiling, Irene is holding a knife.
Mystique and Irene

Claremont had originally intended for Mystique to shift into a man and impregnate Destiny. Thus Mystique and Destiny would have been Nightcrawler's biological parents, however, Marvel and the comics code were having none of that. 

Although she has had other romantic relationships, Destiny is considered Mystique's great love and over time their relationship has been portrayed more explicitly. 

Jennifer Lawrence as Mystique, looking at someone off camera in her natural form.
Image/marvel