The main purpose of the present article is to develop the local functional calculus and to give some applications to local spectral theory. In order to express the results, assume that \(T\in B(X)\) has the single-valued extension property (SVEP) and let \(f\) be a holomorphic function on a neighborhood of a fixed compact subset \(K\) of \(\mathbb{C}\). For \(x\in X\) such that the local spectrum of \(T\) at \(x\), denoted by \(\sigma(x,T)\), is contained in \(K\), define \(f[T]x\) as \[ f[T]x= {1\over 2\pi i} \int_\Gamma f(z) \psi_T(z)\,dz, \] where \(\Gamma\) is an admissible contour and \(\psi_T\) is the local resolvent function. The authors give sufficient conditions for continuity and closability of \(f[T]\), since the operator \(f[T]\) is neither continuous nor closed in general. They study the relation between \(\sigma(x,T)\) and \(\sigma(f[T]x,T)\) and prove a local spectral mapping theorem for a certain class of functions and operators, which asserts that \(\sigma(x, f[T])= f(\sigma(x, T))\) for all \(x\) in the domain of \(f[T]\). As an application of this result, they obtain a stability theorem for the SVEP. Finally, the authors apply a representation of the local resolvent function using the local functional calculus to obtain local resolvent equations.