This work is focused on constructing space-time covariance functions through a hierarchical mixture approach that can serve as building blocks for capturing complex dependency structures. This hierarchical mixture approach provides a unified modeling framework that not only constructs a new class of asymmetric space-time covariance functions with closed-form expressions, but also provides corresponding space-time process representations, which further unify constructions for many existing space-time covariance models. This hierarchical mixture framework decomposes the complexity of model specification at different levels of hierarchy, for which parsimonious covariance models can be specified with simple mixing measures to yield flexible properties and closed-form derivation. A characterization theorem is provided for the hierarchical mixture approach on how the mixing measures determine the statistical properties of covariance functions. Several new covariance models resulting from this hierarchical mixture approach are discussed in terms of their practical usefulness. A theorem is also provided to construct a general class of valid asymmetric space-time covariance functions with arbitrary and possibly different degrees of smoothness in space and in time and flexible long-range dependence. The proposed covariance class also bridges a theoretical gap in using the Lagrangian reference framework. The superior performance of several new parsimonious covariance models over existing models is verified with the well-known Irish wind data and the U.S. air temperature data.