  The present article is a review of recent developments concerning the notion
of F{\o}lner sequences both in operator theory and operator algebras. We also
give a new direct proof that any essentially normal operator has an increasing
F{\o}lner sequence $\{P_n\}$ of non-zero finite rank projections that strongly
converges to 1. The proof is based on Brown-Douglas-Fillmore theory. We use
F{\o}lner sequences to analyze the class of finite operators introduced by
Williams in 1970. In the second part of this article we examine a procedure of
approximating any amenable trace on a unital and separable C*-algebra by
tracial states $\mathrm{Tr}(\cdot P_n)/\mathrm{Tr}(P_n)$ corresponding to a
F{\o}lner sequence and apply this method to improve spectral approximation
results due to Arveson and B\'edos. The article concludes with the analysis of
C*-algebras admitting a non-degenerate representation which has a F{\o}lner
sequence or, equivalently, an amenable trace. We give an abstract
characterization of these algebras in terms of unital completely positive maps
and define F{\o}lner C*-algebras as those unital separable C*-algebras that
satisfy these equivalent conditions. This is analogous to Voiculescu's abstract
characterization of quasidiagonal C*-algebras.
