Frequentist and Bayesian approaches to scientific inference in animal breeding are discussed. Routine methods in animal breeding (selection index, BLUP, ML, REML) are presented under the hypotheses of both schools of inference, and their properties are examined in both cases. The Bayesian approach is discussed in cases in which prior information is available, prior information is available under certain hypotheses, prior information is vague, and there is no prior information. Bayesian prediction of genetic values and genetic parameters are presented. Finally, the frequentist and Bayesian approaches are compared from a theoretical and a practical point of view. Some problems for which Bayesian methods can be particularly useful are discussed. Both Bayesian and frequentist schools of inference are established, and now neither of them has operational difficulties, with the exception of some complex cases. There is software available to analyze a large variety of problems from either point of view. The choice of one school or the other should be related to whether there are solutions in one school that the other does not offer, to how easily the problems are solved, and to how comfortable scientists feel with the way they convey their results.