In these lectures I will discuss the properties of functionals of one-dimensional Brownian motion. These lectures will aim to provide a pedagogical introduction to the path integral methods leading to the derivation of the celebrated Feynman-Kac formula.

The usefulness of this technique in calculating the statistical properties of Brownian functionals will be illustrated with several examples in physics and probability theory, with particular emphasis on applications in computer science. The statistical properties of “first-passage Brownian functionals” and their applications will also be discussed.