  Calabi-Yau (CY) manifolds play a ubiquitous role in string theory. As a
supersymmetry-preserving choice for the 6 extra compact dimensions of
superstring compactifications, these spaces provide an arena in which to
explore the rich interplay between physics and geometry. These lectures will
focus on compact CY manifolds and the long standing problem of determining
their Ricci flat metrics. Despite powerful existence theorems, no analytic
expressions for these metrics are known. In this lecture series we review
numerical approximation methods for Ricci flat CY metrics. Our first aim is to
give a brief overview of the mathematical framework underlying CY geometry, and
the various metrics that CY manifolds admit. We will then discuss the three
types of numerical methods that have been developed to compute Ricci-flat CY
metrics: Donaldson's algorithm, functional minimization methods, and machine
learning methods. Due to the limited time/space we have, this will not be a
comprehensive review, but instead we hope to give a brief survey and illustrate
the essential tools, key ideas, and implementations of this rapidly advancing
field.
