The PnP problem is a classical problem in computer vision, photogrammetry, and even in mathematics. The P5P problem is systematically investigated in this paper. It is proved algebraically that if no 3 control points among the 5 ones are collinear, the P5P problem could have at most 2 solutions, and this upper bound is also attainable. In addition, algebraic conditions are provided for the case of the unique solution and that of the two solutions of the problem respectively and a practical algorithm to compute the admissible solutions also presented. The obtained results are of practical importance in applications such as object pose estimation and robot navigation.