Functions of matrices (more generally, operators) have long attracted the interest of mathematicians and arise frequently in physics  and other fields.  An interesting property of (smooth) functions of large and sparse matrices is that they tend to be strongly localized, i.e, most of the non-negligible entries are concentrated in certain locations; for example, if A is a banded Hermitian matrix, the entries of exp(A) decay super-exponentially in magnitude moving away from the main diagonal.   This property is shared to some extent by more general matrix types and functions, with the precise rate of decay depending on the regularity of the function and on the distance between possible singularities and the spectrum (or numerical range) of the matrix. In my talk I will give an account of recent results on localization for matrix functions and describe some applications of the theory.