High energy resolution studies of 113Cd have been performed with the (d,p) and (d,t) reactions, using polarized beams. In both reactions, a large number of levels (about 80) have been observed up to 2.6 MeV excitation energy, for many of them unambiguous spin and parity assignment being made. Together with previous data from other experiments, the level scheme has probably become essentially complete up to this energy. A detailed comparison is made between the experimental levels and calculations performed with the interacting boson-fermion model-1 (IBFM-1) and with the quasiparticle phonon model (QPM), which allows a good understanding of the level scheme up to about 2 MeV excitation. This nucleus is a special case for the QPM, where due to a strong anharmonicity rather complex configurations must be taken into account in order to get a good description even at low excitation energies. The multiplet structures arising from the coupling of the 3s1/2, 2d3/2, 2d5/2, and 1g7/2 neutron orbitals to the quadrupole one-phonon excitation of the core nucleus 112Cd have been assigned, and the possible identification of 1/2+ and 3/2+ #intruder# states (based on two particle-two hole excitations of the core) is discussed. The observation of many l=1 transitions above 2 MeV excitation gives a clue to the way by which the 11/2- isomeric state can be populated in the (n,[gamma]) and ([gamma],[gamma]') reactions.