$\lambda$-topologies on function spaces

This paper is devoted to the spaces $C_{\lambda}(X)$ of all continuous real-valued functions on $X$ endowed with arbitrary $\lambda$-topologies. This is a fairly complete survey of the results obtained by the author in the following domains of the theory of $\lambda$-topologies: cardinal functions; locally convex properties; weak and strong topologies; dual spaces; lattices of $\lambda$-topologies; completeness.