Abstract: The Brownian excursion measure is a natural measure on boundary-to-boundary curves in a planar domain which has many applications to conformal geometry both deterministic and random. In its simplest form for given starting and ending points, it realises the kernel of the Dirichlet-to-Neumann map for the Laplacian as the total mass of a measure on Brownian paths between those boundary points. In this talk, I will discuss the use of Brownian excursion measures with various boundary conditions to solve problems in the theory of conformal restriction and more recent work (joint with G. Lawler) in which such measures assist in obtaining uniform escape estimates for Schramm--Loewner evolution curves. If time permits I will also discuss the spatial decomposition of conformally invariant random curves.