Recall that a (hyper)graph is -degenerate if each of its nonempty subgraphs has a vertex of degree at most . Every -degenerate (hyper)graph is (easily) -colorable. A (hyper)graph is almost-degenerate if it is not -degenerate, but each of its proper subgraphs is -degenerate. In particular, if is almost -degenerate, then after deleting any edge it is -colorable. For , we study properties of almost -degenerate (hyper)graphs that are not -colorable. By definition, each such (hyper)graph is -critical.