We design and analyze algorithms for the following two robot navigation problems: 1. TARGET SEARCH. Given a robot located at a point s in the plane, how will a robot navigate to a goal t in the presence of unknown obstacles ? 2. LOCALIZATION. A robot is "lost" in an environment with a map of its surroundings. How will it find its true location by traveling the minimum distance ? Since efficient algorithms for these two problems will make a robot completely autonomous, they have held the interest of both robotics and computer science communities. Previous work has focussed mainly on designing competitive algorithms where the robot's performance is compared to that of an omniscient adversary. For example, a competitive algorithm for target search will compare the distance traveled by the robot with the shortest path from s to t. We analyze these problems from the worst-case perspective, which, in our view, is a more appropriate measure. Our results are : 1. For target search, we analyze an algorithm called Dynamic A*. The robot continuously moves to the goal on the shortest path which it recomputes on the discovery of obstacles. A variant of this algorithm has been employed in Mars Rover prototypes. We show that D* takes O(n log n) time on planar graphs and also show a comparable bound on arbitrary graphs. Thus, our results show that D* combines the optimistic possibility of reaching the goal very soon while competing with depth-first search within a logarithmic factor. 2. For the localization problem, worst-case analysis compares the performance of the robot with the optimal decision tree over the set of possible locations. No approximation algorithm has been known. We give a polylogarithmic approximation algorithm and also show a near-tight lower bound for the grid graphs commonly used in practice. The key idea is to plan travel on a "majority-rule map" which eliminates uncertainty and permits a link to the half-Group Steiner problem. We also extend the problem to polygonal maps by discretizing the domain using novel geometric techniques.