We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential $varphi$ is computed by considering only those $(n,epsilon)$-separated sets whose statistical sums with respect to an $m$-dimensional potential $Phi$ are close to a given value $win bR^m$. We then establish for several classes of systems and potentials $varphi$ and $Phi$ a local version of the variational principle. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and Holder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, ergodic localized equilibrium states for Holder continuous potentials are in general not unique.
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