Hybrid simulations with kinetic ions and massless fluid electrons are used to investigate the interaction of collisionless shocks. The interaction between pairs of sub‐ and supercritical shocks are studied for a wide range of parameters relevant to both the solar and interplanetary plasma. These parameters include the shock Mach number, the electron and ion plasma betas, and the angle between the shock normal and the magnetic field. In the collision of two subcritical quasiperpendicular shocks, the shock electromagnetic field structures pass through each other, but do not carry the plasma associated with them. When unequal shocks collide, there is no evidence of the contact discontinuity predicted by fluid theory. In the collision of supercritical quasiperpendicular shocks, significant acceleration of ions occurs. This acceleration arises when ions reflected at one shock encounter the electric fields of the other shock and takes place in a number of distinct phases. For a wide range of parameters, the maximum ion energy is roughly an order of magnitude larger than the ram kinetic energy of the colliding shocks. This ion energization will subsequently manifest itself as a high ion to electron temperature ratio in the doubly shocked plasma. Finally, it was found that the values of the Mach number and angle between the shock normal and magnetic field play a significant role in the collisions. For shock geometries significantly different from perpendicular, ions from each oncoming shock can escape upstream and undergo considerable acceleration.