Abstract: Banach space theorists will certainly tell you many interesting things about L^p spaces ("there is life _in_ L^p"). We exhibit what we believe to be interesting examples of operator algebras _on_ L^p spaces. We will concentrate on L^p analogs of Cuntz algebras and UHF algebras. (There is also work on reduced group L^p operator algebras, crossed products, and L^p analogs of irrational rotation algebras.) The basic properties of the C*-algebras carry over: for the L^p analogs of Cuntz algebras, simplicity, uniqueness, pure infiniteness, amenability, and the K-theory; for the L^p analogs of UHF algebras, simplicity, unique tracial state, and, for fixed p, classification in terms of K-theory. Sometimes the proofs are harder; occasionally, they are completely different (and do not work for p = 2). There are also new phemonena: none of these algebras for one value of p can ever be isomorphic to any such algebra for any other value of p.