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This paper proposes, implements, and evaluates a reinforcement learning (RL)-based computational framework for automatic mesh generation. Mesh generation plays a fundamental role in numerical simulations in the area of computer aided design and engineering (CAD/E). It is identified as one of the critical issues in the NASA CFD Vision 2030 Study. Existing mesh generation methods suffer from high computational complexity, low mesh quality in complex geometries, and speed limitations. These methods and tools, including commercial software packages, are typically semiautomatic and they need inputs or help from human experts. By formulating the mesh generation as a Markov decision process (MDP) problem, we are able to use a state-of-the-art reinforcement learning (RL) algorithm called "soft actor-critic" to automatically learn from trials the policy of actions for mesh generation. The implementation of this RL algorithm for mesh generation allows us to build a fully automatic mesh generation system without human intervention and any extra clean-up operations, which fills the gap in the existing mesh generation tools. In the experiments to compare with two representative commercial software packages, our system demonstrates promising performance with respect to scalability, generalizability, and effectiveness.