As a teenager, Roy learned alchemy from Riza's father, Berthold Hawkeye, and Riza met Roy through this apprenticeship. After Berthold died, Roy convinced Riza to join the military. Together, Mustang and Hawkeye were outstanding soldiers in the Ishval Civil War, though the experience left both of them emotionally scarred.
Later in their military careers, Mustang and Hawkeye were sent to investigate a potential State Alchemist, Edward Elric. Upon arriving at the Elric brothers' home, the two officers discovered that the boys attempted (and failed) the taboo of human transmutation. Mustang and Hawkeye agreed to protect Ed and Al's secret so long as Edward joined the military as a State Alchemist.
For much of the anime/manga, Mustang is a colonel in the Amestrian State Military and Hawkeye is his First Lieutenant. Hawkeye serves as Mustang's most trusted confidant, assistant, and even bodyguard. Hawkeye adopts a stern demeanor around Mustang, scolding him often and making sure he does his necessary paperwork.
As Ed's military superiors, Mustang and Hawkeye keep an eye on the Fullmetal Alchemist and his brother throughout the series. Their protectiveness of the brothers often ropes them into high-stakes situations, including intense battles with supernatural beings. In one such battle, the Homunculus Lust appears to inflict a fatal wound on Mustang. When Lust tells Hawkeye this, Riza completely loses her cool demeanor that she so often wears. She opens fire on Lust, screaming at her and emptying bullet after bullet into the Homunculus's body as tears well up in her eyes. However, it turns out that Mustang is still alive, and he saves Riza from being killed by Lust. Riza collects herself quickly, reverting back to her normal habit of gently scolding Roy. This situation is reversed later on in the series: this time, Riza is in mortal danger and Roy loses his cool. Riza lies in front of a captured Roy, bleeding out on the ground, and Roy is confronted with a terrible choice in order to save her. Roy struggles against his captors, even screaming, "I'll kill you!" at them as they continue to let Riza bleed. To further agitate Mustang, one of them sneers, "A woman precious to you is dying." Fortunately, Roy is able to reach Riza in time, and he embraces her after her wound is closed.
Near the finale of the series, Mustang is blinded as a consequence of being forced to (unwillingly) perform human transmutation. Hawkeye is upset by this, but stands by his side nonetheless. She proves invaluable to Mustang in the final battle against Father as she directs Roy where to send his flames and gives him positive reinforcement.
The epilogue shows an image of Roy and Riza in the future, still in the military and still devoted to one another.
While it's never explicitly stated that Roy and Riza have a romantic interest in each other, there are moments in the manga/anime that suggest that this might be the case. This ambiguity gives Royai shippers the freedom to interpret their relationship in a variety of ways.
Royai fan works often depict Riza and Roy caring for each other on a more intimate level than they are shown to do in canon. Since Riza frequently teases Roy that he's "useless on rainy days" (because water extinguishes his flame alchemy), Royai art and fiction likes to portray the lieutenant tending to a damp and disgruntled Mustang.
- Hiromu Arakawa, the author of Fullmetal Alchemist, was asked by a fan whether or not Roy and Riza got married after the series ended. She replied: "[I] can't get them to marry because of Military Regulations. If they got married, they could no longer stay as superior officer and subordinate." This response could imply that Arakawa thinks of their relationship as romantic.
- Mustang thinks of his subordinate officers as different chess pieces. He believes that Riza is the "Queen" piece, and even calls her "my Queen."
- "Ai," seen at the end of "Royai," also means "love/affection" in Japanese.
- Roy has given Riza permission to shoot him if he ever "strays from the path" and betrays his morals.