  A systematic study is made, for an arbitrary finite relational language with
at least one symbol of arity at least 2, of classes of nonrigid finite
structures. The well known results that almost all finite structures are rigid
and that the class of finite structures has a zero-one law are, in the present
context, the first layer in a hierarchy of classes of finite structures with
increasingly more complex automorphism groups. Such a hierarchy can be defined
in more than one way. For example, the $k$th level of the hierarchy can consist
of all structures having at least $k$ elements which are moved by some
automorphism. Or we can consider, for any finite group $G$, all finite
structures $\mathcal{M}$ such that $G$ is a subgroup of the group of
autmorphisms of $\mathcal{M}$; in this case the "hierarchy" is a partial order.
In both cases, as well as variants of them, each "level" satisfies a logical
limit law, but not a zero-one law (unless $k = 0$ or $G$ is trivial). Moreover,
the number of (labelled or unlabelled) $n$-element structures in one place of
the hierarchy divided by the number of $n$-element structures in another place
always converges to a rational number or to $\infty$ as $n \to \infty$. All
instances of the respective result are proved by an essentially uniform
argument.
