This paper is an outstanding contribution to the analysis of the proximal point algorithm for the solution of the equation 0\(\in Tz\), T a maximal monotone mapping in a Hilbert space H. Let \(P_ k=(I+c_ kT)^{-1}\) be the ”proximal” mapping, \(c_ k>0\), then the algorithm generates the sequence \(Z^{k+1}=P_ kZ^ k\) starting from any \(Z^ 0\in H\). The \(Z^{k+1}\) is approximately computed by the criterion \[ | Z^{k+1}-P_ kZ^ k| \leq \epsilon_ k\min \{1,| Z^{k+1}- Z^ k|^ r\} \] where \(r\geq 0\), \(\epsilon_ k\geq 0\), \(\sum^{\infty}_{k=0}\epsilon_ k<+\infty\). If the set of solutions \(\bar Z=\{z\in H;0\in Tz\}\) is nonempty, the author gives sufficient conditions for linear, superlinear or sublinear convergence of the algorithm involving various assumptions on the coefficients \(c_ k\) and on the growth properties of \(T^{-1}\) around \(\bar Z\). The latter are in the form: There exist \(\delta>0\) such that for all \(w\in B(0,\delta)\) and all \(z\in T^{-1}w\) it follows that \(| z-\bar Z| \leq f(| w|)\) with \(f:[0,+\infty)\to [0,+\infty)\) continuous and \(f(0)=0\). The sharpness of the given convergence rate as well as comparison with previously obtained results and applications are discussed by means of several enlightening examples.