The logo of the university says "The Ohio State University" in white text on a scarlet, gray, or black background:

The Block O has been one of the primary symbols of Ohio State sports for many years. It can be either scarlet or gray, and if a background is included, it will be either scarlet on gray or gray on scarlet, or either on black (here are a couple examples):

A variation on the Block O includes a Buckeye leaf in the lower right-hand corner:

The official logo of the Athletic Department is the Athletic O, which apparently came into use in the early 90s. It is a block O with "Ohio State" written across it. The Block O can be scarlet or gray:

The logo for the Ohio State Alumni Association is a skinny scarlet Block O with a Buckeye leaf in the upper left-hand corner; it can be called the Alumni O: