Most environmental data sets contain measurements collected over space and time. It is the purpose of spatiotemporal statistical models to adequately describe the underlying uncertain spatially explicit phenomena evolving over time. In this talk I will present a new class of spatiotemporal statistical models which is based on stochastic partial differential equations (SPDEs) involving fractional powers of parabolic operators. By means of semigroup theory the corresponding solution processes are rigorously defined, and their spatial and temporal regularity can be quantified. These regularity results provide a key motivation for employing this class of SPDEs in statistical applications: Namely, spatial and temporal smoothness are controlled via two positive parameters, which may be estimated from data in statistical inference. Besides this property, I will discuss further modeling advantages including the long-time behavior and marginal covariance structures.