Living tissues undergo wetting transitions: On a surface, they can either form a dropletlike cell aggregate or spread as a monolayer of migrating cells. Tissue wetting depends not only on the chemical but also on the mechanical properties of the substrate. Here, we study the role of substrate stiffness in tissue spreading, which we describe by means of an active polar fluid model. Taking into account that cells exert larger active traction forces on stiffer substrates, we predict a tissue wetting transition at a critical substrate stiffness that decreases with tissue size. On substrates with a stiffness gradient, we find that the tissue spreads faster on the stiffer side. Furthermore, we show that the tissue can wet the substrate on the stiffer side while dewetting from the softer side. We also show that, by means of viscous forces transmitted across the tissue, the stiffer-side interface can transiently drag the softer-side interface toward increasing stiffness, against its spreading tendency. These two effects result in directed tissue migration up the stiffness gradient. This phenomenon-tissue durotaxis-can thus emerge both from dewetting on the soft side and from hydrodynamic interactions between the tissue interfaces. Overall, our work unveils mechanisms whereby substrate stiffness impacts the collective migration and the active wetting properties of living tissues, which are relevant in development, regeneration, and cancer.