Generating dexterous grasping has been a long-standing and challenging robotic task. Despite recent progress, existing methods primarily suffer from two issues. First, most prior arts focus on a specific type of robot hand, lacking the generalizable capability of handling unseen ones. Second, prior arts oftentimes fail to rapidly generate diverse grasps with a high success rate. To jointly tackle these challenges with a unified solution, we propose GenDexGrasp, a novel hand-agnostic grasping algorithm for generalizable grasping. GenDexGrasp is trained on our proposed large-scale multi-hand grasping dataset MultiDex synthesized with force closure optimization. By leveraging the contact map as a hand-agnostic intermediate representation, GenDexGrasp efficiently generates diverse and plausible grasping poses with a high success rate and can transfer among diverse multi-fingered robotic hands. Compared with previous methods, GenDexGrasp achieves a three-way trade-off among success rate, inference speed, and diversity.