Flight for the player is not present in vanilla Doom and Strife, but implemented in Heretic and Hexen. ZDoom provides flight with the fly cheat and with the flight power granted by items such as the Wings of Wrath.
While flying, a player's forward and backward movements take the free look pitch into account: moving forward while looking up will cause the player to also move upward. This can be used to scale walls, as the horizontal movement is blocked but not the vertical movement.
The commands +moveup and +movedown can be used for pure vertical movement, regardless of view pitch. In addition, jumping will also cause the player to move upwards while flying or swimming. The land command interrupts flight. To resume flying, use +moveup (+jump will not resume flight).