Summary: This work considers a number of properties of space-time covariance functions and how these relate to the spatial-temporal interactions of the process. First, it examines how the smoothness away from the origin of a space-time covariance function affects, for example, temporal correlations of spatial differences. Models that are not smoother away from the origin than they are at the origin, such as separable models, have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances. Smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments. These results are used to obtain a parametric class of spectral densities whose corresponding space-time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time. Second, this work considers models that are asymmetric in space-time; the covariance between site $x$ at time $t$ and site $y$ at time $s$ is different than the covariance between site $x$ at time $s$ and site $y$ at time $t$. A general approach is described for generating asymmetric models from symmetric models by taking derivatives. Finally, the implications of a Markov assumption in time on space-time covariance functions for Gaussian processes are examined, and an explicit characterization of all such continuous covariance functions is given. Several of the new models described in this work are applied to wind data from Ireland.