  This paper has two goals: to present some new results that are necessary for
further study and applications of quasi-linear functionals, and, by combining
known and new results, to serve as a convenient single source for anyone
interested in quasi-linear functionals on locally compact non-compact spaces or
on compact spaces. We study signed and positive quasi-linear functionals paying
close attention to singly generated subalgebras. The paper gives representation
theorems for quasi-linear functionals on $C_c(X)$ and for bounded quasi-linear
functionals on $C_0(X)$ on a locally compact space, and for quasi-linear
functionals on $C(X)$ on a compact space. There is an order-preserving
bijection between quasi-linear functionals and compact-finite topological
measures, which is also "isometric" when topological measures are finite.
Finally, we further study properties of quasi-linear functionals and give an
explicit example of a quasi-linear functional.
