  The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF)
on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity
conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset
$S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb
R$ we construct a convex set of curvature operators $C(S,F)$, which is
invariant under the HCF. Varying $S$ and $F$, we prove that the HCF preserves
Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic
orthogonal bisectional curvature, lower bounds on the second scalar curvature.
As an application, we prove that periodic solutions to the HCF can exist only
on manifolds $M$ with the trivial canonical bundle on the universal cover
$\widetilde{M}$.
