<p>In this paper, we consider a priori estimates and examine the existence of bounded solutions for a class of systems of ordinary differential equations whose right-hand sides have an arbitrary order of growth with respect to the independent and dependent variables. For the right-hand sides of the equations, we find one-sided estimates that provide a priori estimates for bounded solutions; using the methods for calculating the rotation of vector fields and the method of periodic truncations, we prove theorems on the existence of periodic and bounded solutions.</p>