In this survey we introduce the general Theory of Approximation to functions in (quasisemi)normed spaces. The exposition starts with an explanation of the main problem: we consider a family of subspaces (our approximants), and we obtain a description of the subspace(s) that are approximated by this family with a given approximation order. We also introduce some of the most common tools used to solve these problems.

Approximation Theory gets heavily improved when some efforts are put into the effective construction of the approximants on each given example, rather than simply stating its existence—this is what we call Constructive Approximation.

The fact that we can handle actual functions allows us to obtain yet more properties of the approximants. It is implicit throughout the exposition how Approximation Theory benefits from other branches of Mathematics, but also how Constructive Approximation can be used to prove results from those other fields.