Multi-precision arithmetic means floating point arithmetic supporting multiple, possibly arbitrary, precisions. In recent years there has been a growing demand for and use of multi-precision arithmetic in order to deliver a result of the required accuracy at minimal cost. For a rapidly growing body of applications, double precision arithmetic is insufficient to provide results of the required accuracy. These applications include supernova simulations, electromagnetic scattering theory, and computational number theory. On the other hand, it has been argued that for climate modelling and deep learning half precision (about four significant decimal digits) We discuss a number of topics involving multi-precision arithmetic. Lecture includes: (i) How to derive linear algebra algorithms that will run in any precision, as opposed to be being optimized (as some key algorithms are) for double precision. (ii) The need for, availability of, and ways to exploit, higher precision arithmetic (e.g., quadruple precision arithmetic). (iii) What accuracy rounding error bounds can guarantee for large problems solved in low precision. (iv) How a new form of preconditioned iterative refinement can be used to solve very ill conditioned sparse linear systems to high accuracy.