|Efficient trajectory planning algorithms are a crucial issue for modern autonomous underwater vehicles. Classical trajectory planning algorithms in artificial intelligence are not designed to deal with wide continuous environ~ents prone to currents. Furthermore torpedo-like underwater vehicles are strongly nonholonomic. A novel Fast Marching based approach is proposed to address the following theoretical issues. First, an algorithm called FM* is developed to efficiently extract a 2D continuous and derivable curve from a discrete representation of the environment. Second, underwater currents are taken into account thanks to an anisotropic extension of the original Fast Marching algorithm. Third, the vehicle turning radius is introduced as a constraint on the curvature of the optimal.trajeCtory for both isotropic and anisotropic media. FUrther developments are proposed to optimize the Fast Marching based method to real-time constraints. On one hand, a fast multiresolution method is introduced to extract suboptimal trajectories. On the other hand, a dynamic version of the Fast Marching algorithm called DFM is developed to efficiently replan trajectories in dynamic unpredictable environments. Besides, it is shown that DFM algorithm is an excellent tool for visibility-based trajectory planning in a-priori unknown domains. The overall Fast Marching based trajectory planning method has been tested on simulated underwater environments and validated on a real experimental platform in open water. Keywords: artificial intelligence, trajectory planning, Fast Marching algorithm, autonomous underwater vehicle, isotropic and anisotropic ordered upwind methods, functional minimization, curvature radius, unknown environment, multiresolution method, dynamic replanning, visibility-based navigation.