  This is a survey of recent progress in several areas of combinatorial
algebra. We consider combinatorial problems about free groups, polynomial
algebras, free associative and Lie algebras. Our main idea is to study
automorphisms and, more generally, homomorphisms of various algebraic systems
by means of their action on ``very small" sets of elements, as opposed to a
traditional approach of studying their action on subsystems (like subgroups,
normal subgroups; subalgebras, ideals, etc.) We will show that there is a lot
that can be said about a homomorphism, given its action on just a single
element, if this element is ``good enough". Then, we consider somewhat bigger
sets of elements, like, for example, automorphic orbits, and study a variety of
interesting problems arising in that framework.
  One more point that we make here is that one can use similar combinatorial
ideas in seemingly distant areas of algebra, like, for example, group theory
and commutative algebra. In particular, we use the same language of
``elementary transformations" in different contexts and show that this approach
appears to be quite fruitful for all the areas involved.
