The replacement of ordinary with finite-part integrals, which generalize the definition originally given by J. Hadamard, was shown previously to yield ipso fucto correctly renormalized field equations independently of perturbative techniques, provided due care is taken of the symmetry properties of the theory and some quantitative conditions (closely related to Dyson's heuristic conditions for renormalizability) are satisfied by the theory itself. The class of finite- part integrals considered before was extended to cover all instances which may practically occur. Several possible prescriptions in configuration space which exhibit the wanted properties are discussed. It appears that some prescriptions are better suited to the study of general questions, such as renormalizability of a theory and deduction of the Lie equations of its renormalization group, while others are more convenient for actual computation, once renormalizability is proved. (auth)