Tantalum arsenide is a member of the noncentrosymmetric monopnictides, which are putative Weyl semimetals. In these materials, three-dimensional chiral massless quasiparticles, the so-called Weyl fermions, are predicted to induce novel quantum mechanical phenomena, such as the chiral anomaly and topological surface states. However, their chirality is only well defined if the Fermi level is close enough to the Weyl points that separate Fermi surface pockets of opposite chirality exist. In this Letter, we present the bulk Fermi surface topology of high quality single crystals of TaAs, as determined by angle-dependent Shubnikov-de Haas and de Haas-van Alphen measurements combined with ab initio band-structure calculations. Quantum oscillations originating from three different types of Fermi surface pockets were found in magnetization, magnetic torque, and magnetoresistance measurements performed in magnetic fields up to 14 T and temperatures down to 1.8 K. Of these Fermi pockets, two are pairs of topologically nontrivial electron pockets around the Weyl points and one is a trivial hole pocket. Unlike the other members of the noncentrosymmetric monopnictides, TaAs is the first Weyl semimetal candidate with the Fermi energy sufficiently close to both types of Weyl points to generate chiral quasiparticles at the Fermi surface.