This chapter covers both linear and multilinear algebra fields. It highlights linear algebra algorithms and challenges on their path to exascale. Scalable numerical tensor algebra algorithms are currently under active development, and their efficient portable implementation in domain-specific libraries is in very high demand. Exascale scientific computing systems will likely be characterized by a diverse set of computer architectures as the T.Moore's law era is approaching its critical point. Threading is recognized as an essential requirement for attaining high performance on pre-exascale and exascale systems. Usage patterns at high-performance computing facilities indicate that dense and sparse linear algebra operations are among the most commonly used in scientific applications. The dense matrix–matrix product operation is core service, which enables high efficiency for many higher level linear algebra operations that depend on it. Extreme scale computing presents an enormous opportunity for breakthrough scientific simulations as well as grand challenge of writing scalable, efficient, portable, and maintainable software for performing these simulations.