  This is a survey on the relation between homological properties of the
Frobenius endomorphism and finiteness of various homological dimensions of the
ring or of modules over it, such as global dimension and projective dimension.
We begin with Kunz's surprising result in 1969 that the regularity of a
Noetherian local ring is equivalent to the flatness of its Frobenius
endomorphism, as well as the subsequent generalizations to the module setting
by Peskine and Szpiro and continue up through the recent flurry of results in
the last five years. An attempt is made to include proofs whenever feasible.
