A nonlinear version of the well-known Peetre theorem on the locally finite order of support nonincreasing linear operators is a useful tool for investigation of possible orders of natural oprators between natural bundles and also for some other purposes. The author presents the analytical aspects of nonlinear generalizations of the Peetre theorem in a rather general setting, gives a counterexample showing that in general the obtained results are the best possible and discusses some concrete geometrical applications.