  In this paper, we establish the curvature estimates for $p$-convex
hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq
\frac{n}{2}$. The existence of a star-shaped hypersurface of prescribed
curvature is obtained. We also prove a type of interior $C^2$ estimates for
solutions to the Dirichlet problem of the corresponding equation.
