  A complete first order theory of a relational signature is called monomorphic
iff all its models are monomorphic (i.e. have all the $n$-element substructures
isomorphic, for each positive integer $n$). We show that a complete theory
${\mathcal T}$ having infinite models is monomorphic iff it has a countable
monomorphic model and confirm the Vaught conjecture for monomorphic theories.
More precisely, we prove that if ${\mathcal T}$ is a complete monomorphic
theory having infinite models, then the number of its non-isomorphic countable
models, $I({\mathcal T},\omega )$, is either equal to $1$ or to ${\mathfrak
c}$. In addition, $I({\mathcal T},\omega)= 1$ iff some countable model of
${\mathcal T}$ is simply definable by an $\omega$-categorical linear order on
its domain.
