A theory of model choice in regression analysis is developed, covering tho problems of approximation of regression functions with unknown functional form, of optimal prediction for the realization of some dependend variables, of polynomial and multiple regression. In this (decision theoretical) frame a survey of the known model choice procedures including stepwise procedures and their variants is given. Moreover, several new procedures and variants are described, e.g. BAYEsian, minimax and empirical model choice, global and robust backward elimination or stepwise regression. The procedures of “maximal multiple correlation”“optimal regression” and the C_p-criterion of MALLOWS are obtained as special cases of ε-BAYES, BAYES and empirical model choice respectively.Some comparisons of procedures considering the risk function are reported, e.g. under some assumptions the lower global variant of forward selection is better than the choice of the largest model.The parameters of this procedure can be chosen in such a way, that a very small model. The parameters of this procedures can be chosen in such a way, that a very small model is chosen, that approximating an unknown regression function by a linear combination of few given functions.