LetW=(W t, t≧0) denote a two-dimensional Brownian motion starting at 0 and, for 0<α<π, letC α be a wedge in ℝ2 with vertex 0 and angle 2α. We consider the set of timest's such that the path ofW, up to timet, stays inside the translated wedgeW t-Cα. It follows from recent results of Burdzy and Shimura that this set, which we denote byH α, contains nonzero times if, and only if, α>π/4. Here we construct a measure, a local time, supported onH α. For π/4<α≦α/2, the Brownian motionW, time-changed by the inverse of this local time, is shown to be a two-dimensional stable process with index 2-π/2α. This results extends Spitzer's construction of the Cauchy process, which is recovered by taking α=π/2. A formula which describes the behaviour ofW before a timet∈H α is established and applied to the proof of a conjecture of Burdzy. We also obtain a two-dimensional version of the famous theorem of Levy concerning the maximum process of linear Brownian motion. Precisely, for 0<α<π/2, letS t denote the vertex of the smallest wedge of the typez-C α which contains the path ofW up to timet. The processS t-Wt is shown to be a reflected Brownian motion in the wedgeC α, with oblique reflection on the sides. Finally, we investigate various extensions of the previous results to Brownian motion inR d, d≧3. LetC Ω be the cone associated with an open subset Ω of the sphereS d-1, and letH Ω be defined asH α above. Sufficient conditions are given forH Ω to contain nonzero times, in terms of the first eigenvalue of the Dirichlet Laplacian on Ω.