Before we had first-person shooters, we had side scrolling shooters, also known run & gun shooters. Side-scrolling shooters are usually action games that involved you and your gun progressing horizontally across a level. These games were extremely popular in the days of the Genesis and SNES, before 3D became the standard and the mainstream decided 2D was outdated.
Side scrolling shooters are still made today, and not just on the Game Boy Advance or Nintendo DS. Although rare, developers still create side scrolling shooters for home consoles, such as Contra: Shattered Soldier.
Examples of Side Scrolling Shooters