  We study uncountable structures similar to the Fra\"iss\'e limits. The
standard inductive arguments from the Fra\"iss\'e theory are replaced by
forcing, so the structures we obtain are highly sensitive to the universe of
set theory. In particular, the generic structures we investigate exist only in
generic extensions of the universe. We prove that in most of the interesting
cases the uncountable generic structures are rigid. Moreover, we provide a
(consistent) example of an uncountable, dense set of reals with the group of
integers as its automorphism group.
