We develop a logic which is a variant of continuous logics introduced by Chang and Keisler in 1966. We show that it is a common generalization of classical first order logic as well as Henson's logic for Banach spaces, and many good properties of first order logic (such as compactness) carry over. Continuous logic can axiomatize natural classes arising in functional analysis, dynamics, the theory of representations of locally compact groups, and is a suitable and convenient framework for model-theoretic study of classes of metric spaces. While classical model theory is applied mostly to algebraic structures, our framework pushes out the boundary in which logic can be used to a much broader context of a metric space equipped with continuous extra-structure.