As one of the independent control parameters, pressure plays an important role in finding new phenomena, testing related theories and guiding the explorations for new superconductors. In this review article, we will briefly review the progress achieved from high pressure studies on some main types of the iron pnictide superconductors, including 1111-type, 122-type, 111-type, 10-3-8 type and 112-type. A few typical results from high pressure studies are introduced in more detail, including the positive pressure effect on the superconducting transition temperature TC of 1111-type iron pnictide superconductors, which indicates a way to enhance the TC by using a smaller cation to replay La ion; the maximum TC of iron pnictide superconductors estimated by high-pressure studies on a series of 1111-type iron-based superconductors etc. More importantly, high pressure studies on the parent compounds of iron pnictide superconductors clearly demonstrate that pressure can suppress the transition temperatures of magnetic order and crystal structure, and then drive a superconducting transition. Furthermore, many examples are given in this review to reveal how the magnetic order competes with superconductivity under pressure, which provides new constrains for the establishment of the theory on superconductivity. These high pressure results are expected to be helpful for the studies of high-TC superconductors and for the exploring of new superconductors.