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.