The center has multiple responsibilities, but catching passes isn't one of them.
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Football is a heavily regulated and officiated sport. Every possible play has many strict rules on which player can do what actions, and which actions are prohibited. Often, these rules are designed to prevent plays that are too tricky or deceiving, giving a defense a fair chance of equitable competition. For example, the rules clearly detail who can and cannot catch or run the ball after the snap for a variety of different situations. The center, generally, cannot touch the ball again after the snap, except for certain situations.
The center is the only player on the field, on either offense or defense, who touches the ball before the snap. The rules deny the center the ability from simply picking up the ball and lunging forward with it, as that would be too difficult for a defense to stop. The center must snap the ball to another player, either the quarterback or a running back. Once the snap takes place, the center engages with the defensive line to block defensive linemen and clear space for the play.
Eligible and Ineligible Receivers
A tackle or guard on the offensive line can, by rule, catch the ball as long as he declares himself an eligible receiver, lines up at either end of the offensive line, or lines up at least 1 yard behind the line of scrimmage. To do this, the lineman tells the referee that he is an eligible receiver, and in most cases the defense becomes aware of this declaration. One lineman per play, per formation may declare as an eligible receiver. The center cannot declare as an eligible receiver while also being the player who snaps the ball, and being unable to line up in a permissible way.
Tipped or Batted Balls
The center can catch a ball if it is tipped or batted at by any defensive player, or by any receiver or running back on his own team. Once the ball makes contact with a defender or eligible player, any other player on the field can legally catch it and advance the play without penalty. Typically, plays where a forward pass is deflected by the outstretched hand of the defender so that it pops straight in the air are the most common times a center legally catches it.
The other exception to the general prohibition on centers catching the ball is during lateral passes. As long as the quarterback is not throwing a forward pass directly to a player, any other player is eligible to touch the ball. This means that the ball cannot be thrown forward at all, not even a single yard. It must be thrown to one side of the quarterback, or behind him. For example, the quarterback pitches the ball to a running back, who attempts to run upfield. Seeing that he's blocked, he pitches the ball sideways to another player, who then pitches it behind him to the center just to keep the play alive. This is a legally permissible catch.