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This paper characterizes time-optimal trajectories in anisotropic (direction-dependent) environments where path curvatures are bounded by the inverse of the minimum-turning radius of a mobile agent. Such problems are often faced in the navigation of aerial, ground and naval vehicles when a mobile agent cannot instantaneously change its heading angle. The work presented is a generalization of the Dubins car problem, which considers the fastest paths with bounded curvature while assuming constant speed and minimum-turning radius. We relax this assumption and discuss fastest-path finding problems for the generalized direction-dependent speed and minimum-turning radius functions, to account for the effects of waves, winds and slope of the terrain on the agent’s motions. We establish that there exists an optimal path such that it is a portion of a path of the type [Formula: see text] where [Formula: see text] denotes a sharpest-turn curve and [Formula: see text] a straight-line segment. We further analyze a special case wherein the speed polar plot is convex, and show that in that case there exists an optimal path with the same structure as for the Dubins problem: [Formula: see text] or [Formula: see text]. An algorithm that implements our results for the convex speed polar plot is also presented.