Let G be a p-adic reductive group with Lie algebra g. In this talk, we'll briefly review Schneider and Teitelbaum's theory of locally analytic representations of G and then we'll discuss a functor which constructs locally analytic representations of G out of g-modules in the extension closure of the Bernstein-Gelfand-Gelfand category O. A key role in this construction is played by p-adic logarithms. This construction is joint work with with Matthias Strauch, and generalizes earlier work by Strauch and Orlik.