  In this paper we study the Buchstaber invariant of simplicial complexes,
which comes from toric topology. With each simplicial complex $K$ on $m$
vertices we can associate a moment-angle complex $\mathcal Z_K$ with a
canonical action of the compact torus $T^m$. Then $s(K)$ is the maximal
dimension of a toric subgroup that acts freely on $\mathcal Z_K$. We develop
the Buchstaber invariant theory from the viewpoint of the set of minimal
non-simplices of $K$. It is easy to show that $s(K)=1$ if and only if any two
and any three minimal non-simplices intersect. For $K=\partial P^*$, where $P$
is a simple polytope, this implies that $P$ is a simplex. The case $s(P)=2$ is
such more complicated. For example, for any $k\geqslant 2$ there exists an
$n$-polytope with $n+k$ facets such that $s(P)=2$. Our main result is the
criterion for the Buchstaber invariant of a simplicial complex $K$ to be equal
to two.
