**Two locality properties in two dimensions**

In two dimensions, many self-interacting processes are described by the Schramm-Loewner Evolution SLE(kappa), a family of random fractal path joining two boundary points of an underlying domain D. These continuous paths arise as the scaling limits of various discrete self-interacting paths, such as loop-erased random walk.

A self-interacting process has the locality property if it does not "feel" the boundary of its domain D until it hits the boundary. Among the two-dimensional processes known as Schramm Loewner Evolution SLE(kappa), it is known that only one, SLE(6), satisfies the locality property. In this talk, I will describe the key properties that identify SLE(6) - the Domain Markov Property, conformal invariance, and the (classical) Locality Property - and introduce a "non-local" form of locality also satisfied by SLE(6), describing the behaviour of the process when it first encloses a target set.