Summary: We construct a class of conformally invariant measures on sets (or paths) and we study the critical exponents called intersection exponents associated to these measures. We show that these exponents exist and that they correspond to intersection exponents between planar Brownian motions. More precisely, using the definitions and results of our paper [Ann. Probab. 27, No.4, 1601–1642 (1999; Zbl 0965.60071)], we show that any set defined under such a conformal invariant measure behaves exactly as a pack (containing maybe a non-integer number) of Brownian motions as far as all intersection exponents are concerned. We show how conjectures about exponents for two-dimensional self-avoiding walks and critical percolation clusters can be reinterpreted in terms of conjectures on Brownian exponents.