A simply connected region Ω in the complex plane C with smooth boundary <9Ω is called A>convex (k > 0) if k(z 9 dΩ) > k for all z £ Ω, where k(z,dΩ) denotes the euclidean curvature of dΩ at the point z.A different definition is used when dΩ is not smooth.We present a study of the hyperbolic geometry of /c-convex regions.In particular, we obtain sharp lower bounds for the density A Ω of the hyperbolic metric and sharp information about the euclidean curvature and center of curvature for a hyperbolic geodesic in a /c-convex region.We give applications of these geometric results to the family K(k, a) of all conformal mappings / of the unit disk D onto a /c-convex region and normalized by /(0) = 0 and /'(0) = a > 0. These include precise distortion and covering theorems (the Bloch-Landau constant and the Koebe set) for the family K(k,a).