The purpose of this research is to develop a scalable and highly optimized open source library for hierarchical low-rank approximation of dense matrices. Such large dense matrices naturally appear in electromagnetic, seismic, quantum, and fluid simulations, in scientific computing. Large dense matrices also appear in machine learning, where the Hessian, Fisher, Covariance, and Gram matrices play an important role in determining the properties of optimization and generalization of deep neural networks. Unlike their dense counterparts which require O(N³) time and O(N²) memory. H-matrices can perform matrix multiplication and factorization in O(Nlog²N) time and O(N) memory, have controllable arithmetic intensity, have asynchronous communication, and can exploit deep memory hierarchy, which makes them an ideal solver/preconditioner for the Exascale era.