There is a growing consensus that cyclotron resonances play important roles in heating protons and ions in coronal holes where the fast solar wind originates and throughout interplanetary space as well. Most work on cyclotron resonant interactions has concentrated on the special, but unrealistic, case of propagation along the ambient magnetic field, B0, because of the great simplification it gives. This paper offers a physical discussion of how the cyclotron resonances behave when the waves propagate obliquely to B0. We show how resonances at harmonics of the cyclotron frequency come about, and how the physics can be different depending on whether E⊥ is in or perpendicular to the plane containing k and B0 (k is wave vector, and E⊥ is the component of the wave electric field perpendicular to B0). If E⊥ is in the k-B0 plane, the resonances are analogous to the Landau resonance and arise because the particle tends to stay in phase with the wave during the part of its orbit when it is interacting most strongly with E⊥. If E⊥ is perpendicular to the k-B0 plane, then the resonances depend on the fact that the particle is at different positions during the parts of its orbit when it is interacting most strongly with E⊥. Our main results are our equations (10), (11), and (13) for the secular rate of energy gain (or loss) by a resonant particle and the unfamiliar result that ions can resonate with a purely right-hand circularly polarized wave if the propagation is oblique. We conclude with some speculations about the origin of highly obliquely propagating ion resonant waves in the corona and solar wind. We point out that there are a number of instabilities that may generate such waves locally in the corona and solar wind.