We will survey the various noncommutative versions of dimension for C*-algebras and their applications, beginning with Rieffel's stable rank. Other dimension theories include real rank, tracial rank, decomposition rank, and nuclear dimension, all of which have different applications. Uses of these theories in the structure of approximately homogeneous C*-algebras will be discussed, including approximate divisibility and Z-stability. Finally, we discuss stability questions, the Cuntz semigroup, and radius of comparison.