Summary: Relations between Lagrangian structures, metric structures, and semispray connections on a manifold are investigated. Generalized Finsler structures (called quasifinslerian) are studied, coming from integrable time, position and velocity dependent metrics. For every quasifinslerian metric one has a naturally associated semispray connection, called canonical connection, and a global Lagrangian, called kinetic energy. One obtains the most general form of metrical connections and related equations for geodesics, which at the same time are variational. As expected, canonical connections generalize the Levi--Civita connection and the connections appearing in Finsler geometry. Relations between quasifinslerian and Lagrange spaces, as well as between metrizability of semispray connections and the existence of variational integrators for second-order ordinary differential equations are also discussed.