Carbon nanotubes (CNT) posses a very large application potential in molecular electronic devices. Infinite single-wall metallic CNTs have theoretically a conductance of 2G₀ (G₀=2e2/h) because of the two electronic bands crossing the Fermi level. For finite size CNTs experiments have shown that other values are also possible, indicating a very strong influence of the contact. We study electron transport in single- and double-wall CNTs contacted to metallic electrodes with an fcc(111) surface within the Landauer transport formalism combined with Green function techniques. The electronic structure of the CNTs is described with a simple -orbital model. We show that the symmetry of the contact region may lead in some cases to blocking of one of the transport channels reducing the conductance to 1G₀ for single-wall CNTs. In the case of double-wall systems with both inner and outer shells being metallic, non-diagonal self energy contributions from the electrodes may induce mixing of the CNT's channels, again reducing the conductance from the theoretically expected value (4G₀). Finally, we discuss the influence of inter-wall interactions on the conductance.