This book presents a unified approach to Korovkin-typeapproximation theorems. It includes classical material onthe approximation of real-valuedfunctions as well as recentand new results on set-valued functions and stochasticprocesses, and on weighted approximation. The results arenotonly of qualitative nature, but include quantitativebounds on the order of approximation.The book is addressed to researchers in functional analysisand approximation theory as well as to those that want toapplythese methods in other fields. It is largely self-contained, but the readershould have a solid background inabstract functional analysis.The unified approach is based on a new notion of locallyconvex ordered cones that are not embeddable in vectorspaces but allow Hahn-Banach type separation and extensiontheorems. This concept seems to be of independent interest.