This book presents a unified approach to Korovkin-type approximation theorems. It includes classical material on the approximation of real-valued functions as well as recent and new results on set-valued functions and stochastic processes, and on weighted approximation. The results are not only of qualitative nature, but include quantitative bounds on the order of approximation. The book is addressed to researchers in functional analysis and approximation theory as well as to those that want to apply these methods in other field. It is largely self-contained, but the reader should have a solid background in abstract functional anaysis. The unified approach is based on a new notion of locally convex ordered cones that are not embeddable in vector spaces but allow Hahn-Banach type separation and extension theorems. This concept seems to be of independent interest.