Spatial-temporal data arises in many applications, for example, environment sciences and disease mapping. This dissertation focuses on Gaussian spatial-temporal data. To make statistical inference for Gaussian spatial-temporal data, we developed a special class of spatial-temporal Gaussian state-space models in which the state vectors are constructed following spatial-temporal Gaussian autoregressions that are consistent with the conditional formulation of auto-normal spatial fields. We then proposed a matrix-free h-likelihood method to predict state vectors and to estimate parameters. The proposed method provides the same inference as that obtained from the Kalman filter and residual maximum likelihood analysis. However, for data from a large number of spatial sites, it is shown to have significant computational advantages. The novel elements of the dissertation include how we present a spatial-temporal state-space model as a linear regression model and develop estimation by matrix-free computations. Furthermore, the dissertation details inference in small time steps, indicates how the proposed method can be adapted to other complex spatial-temporal dynamical models based on stochastic partial differential equations, and discusses matrix-free Bayesian computations for non-Gaussian spatial data modeling. The method applies to data with both regularly and irregularly sampled spatial locations, and is illustrated through a simulation study and two data examples, one with monthly soil moistures across North America and the other with atmospheric concentrations of total nitrate across Eastern North America.