  The study of operator algebras on Hilbert spaces, and C*-algebras in
particular, is one of the most active areas within Functional Analysis. A
natural generalization of these is to replace Hilbert spaces (which are
$L^2$-spaces) with $L^p$-spaces, for $p\in [1,\infty)$. The study of such
algebras of operators is notoriously more challenging, due to the lack of
orthogonality in $L^p$-spaces.
  We give a modern overview of a research area whose beginnings can be traced
back to the 50's, and that has seen renewed attention in the last decade
through the infusion of new techniques. The combination of these new ideas with
old tools was the key to answer some long standing questions. Among others, we
provide a description of all unital contractive homomorphisms between algebras
of $p$-pseudofunctions of groups.
