Efficient path-planning algorithms are a crucial issue for modern autonomous underwater vehicles. Classical path-planning algorithms in artificial intelligence are not designed to deal with wide continuous environments prone to currents. We present a novel Fast Marching (FM)-based approach to address the following issues. First, we develop an algorithm we call FM* to efficiently extract a 2-D continuous path from a discrete representation of the environment. Second, we take underwater currents into account thanks to an anisotropic extension of the original FM algorithm. Third, the vehicle turning radius is introduced as a constraint on the optimal path curvature for both isotropic and anisotropic media. Finally, a multiresolution method is introduced to speed up the overall path-planning process.