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This paper presents the prior PAC-Bayes bound and explores its capabilities as a tool to provide tight predictions of SVMs' generalization. The computation of the bound involves estimating a prior of the distribution of classifiers from the available data, and then manipulating this prior in the usual PAC-Bayes generalization bound. We explore two alternatives: to learn the prior from a separate data set, or to consider an expectation prior that does not need this separate data set. The prior PAC-Bayes bound motivates two SVM-like classification algorithms, prior SVM and h- prior SVM, whose regularization term pushes towards the minimization of the prior PAC-Bayes bound. The experimental work illustrates that the new bounds can be significantly tighter than the original PAC-Bayes bound when applied to SVMs, and among them the combination of the prior PAC-Bayes bound and the prior SVM algorithm gives the tightest bound.