Former CIA Chief Jonna Mendez was the head of the agency's department that was involved with disguises for their agents and spies. She goes into depth (and probably barely scratching the surface at the same time for the CIA) about how disguises are used in the field.
There are a couple of obvious things they would do, mostly changing a person's features to the opposite of what they have. So a spy with a beard would shave it, straighten their curly hair and add grey hair and wrinkles while young. The point being the person who they're meeting with writes a description of the spy as the total opposite of what they're actually like.
But then, there are some pretty sophisticated disguises. Mendez herself met with George H.W. Bush back when he was President wearing a full mask and wig, unbeknownst to him. Bush was once the director of the CIA so the disguise must have been pretty damn good. There are also pieces to change the look of you gums and pieces that attach to the roof of your mouth to change your voice in case it is being recorded.
Pretty incredible stuff.