A class of renormalization groups characterized as the general Gaussian renormalization group is defined. It includes the sharp cutoff renormalization group of Wilson, the linear renormalization group of Bell and Wilson, and the incomplete‐integration renormalization group of Wilson. The renormalization group approach to critical phenomena and the expansion around four dimensions is reviewed. The fixed point of the general Gaussian renormalization group is determined to order &egr;2, where &egr;=4−d(dis the dimensionality of the system), and the critical exponents are determined to the same order. The main conclusion is that the exponents do not depend on the cutoff functions of the renormalization group.