Let (M,g) be a compact oriented n-dimensional smooth Riemannian manifold. Consider the following quadratic Riemannian functional$$SR(g) = \int\sb{M}\ \vert R\sb{ijkl}(g)\vert \sp{2}d\mu$$which is homogeneous of degree n2−2, where R\sbijkl(g) is the curvature tensors of (M,g) and dμ is the volume element measured by g. A critical point of SR(g) is called a critical metric on M, that is, the Ricci tensor satisfies the critical equations gradSR\sbg = 0. In particular, for a compact 4-manifold M, every Einstein metric is a critical metric for SR on M. In this thesis, we propose an extension of the compactness property for Einstein metrics to critical metrics on a compact smooth Riemannian 4-manifold M. More precisely, first we consider the subspace G(M) of all critical metrics on M with the injectivity radius bounded from below by a constant i\sb0> 0 and diameter bounded from above by d. Then we are able to prove that G(M) is compact as a subset of moduli space of critical metrics in the C\sp∞-topology (Theorem 6.1). Second, we replaced the injectivity radius lower bound by the local volume bound, then we get a compact 4-dimensional critical orbifold (Theorem 7.1). Furthermore, by using the fundamental equations of Riemannian submersions with totally geodesic fibers, we construct some critical Riemannian 4-manifolds.