This dissertation is about computational tools based on randomized numerical linear algebra for handling larg-scale matrix data. Since large datasets have become commonly available in a wide variety of modern applications, there has been an increasing demand for numerical methods for storing, processing, and learning from them. Matrices, the classical form for representing datasets, naturally connect these tasks with the rich literature of numerical linear algebra. For a diverse collection of problems, randomized methods offer extraordinary efficiency and flexibility. This work focuses on using randomized numerical linear algebra to build practical algorithms for problems of massive size and high complexity that traditional methods are unable to handle. Through this dissertation, we explore topics across network science, Gaussian process regression, natural language processing, and quantum chemistry. Our contribution includes a collection of scalable and robust numerical methods under a unifying theme, accompanied by efficient implementations. As a result, we are able to significantly speed up the computation for several existing applications, and explore problems and datasets that were intractable before.