In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by □, i.e., the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a □-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voevodsky's category of motives and A¹-invariant theories : assuming resolution of singularities, we identify the latter with the full subcategory comprised of A¹-local objects in the category of logarithmic motives. Fundamental properties such as □-homotopy invariance, Mayer-Vietoris for coverings, the analogs of the Gysin sequence and the Thom space isomorphism as well as a blow-up formula and a projective bundle formula witness the robustness of the setup.