Areas of planar Brownian curves

Addresses the problem of the algebraic area enclosed by a Brownian curve in two dimensions, recently reconsidered by Khandekar and Wiegel (1988). The author recalls the principal results actually first obtained by Paul Levy in 1950. Another derivation by functional integrals is given for several probability distributions: that of the area of a Brownian ring, the area between an open Brownian arc and its chord and, finally, the area swept with respect to the mean position of the Brownian path, as a result which seems to be new.