The Kondo effect is usually connected with the interaction between a localized spin moment and itinerant electrons. This interaction leads to the formation of a narrow resonance at the Fermi level, which is called the Abrikosov-Suhl or Kondo resonance. Scanning tunnelling microscopy is an ideal technique for real-space investigations of complicated electronic structures and many-body phenomena, such as the formation of the Kondo resonance or d-wave pairing in high-T(c) superconductors. Theory has predicted that similar, Kondo-like many-electron resonances are possible for scattering centres with orbital instead of spin degrees of freedom--the quadruple momenta in uranium-based compounds or two-level systems in metallic glasses are examples of such 'pseudo-Kondo' scattering centres. Here we present evidence for the orbital Kondo resonance on a transition-metal surface. Investigations of an atomically clean Cr(001) surface at low temperature using scanning tunnelling microscopy reveal a very narrow resonance at 26 meV above the Fermi level, and enable us to visualize the orbital character of the corresponding state. The experimental data, together with many-body calculations, demonstrate that the observed resonance is an orbital Kondo resonance formed by two degenerate d(xz), d(yz) surface states.