Detailed description of the heterogeneity of geologic formations is needed to make accurate predictions of flow and transport in such formations. However, only limited measurements at a few locations are usually available. This combination of significant spatial heterogeneity with incomplete information about it leads to uncertainty about the values of formation properties and thus, to uncertainty in flow and transport prediction. The theory of stochastic processes provides a natural method for quantifying these uncertainties. During the last three decades, many stochastic models have been developed for this purpose. In most of these models, the assumption of stationary (statistically homogeneous) medium properties and flow, among other simplifying ones, is invoked. In the vadose zone or an unsaturated-saturated system, the requirement of flow stationarity is always violated because the presence of water table boundary renders the flow highly nonstationary (local dependent). In this presentation, we describe recently developed nonstationary stochastic models for flow and solute transport and discuss the impacts of flow nonstationarity on uncertainty quantification for the following scenarios: the gravity-dominated flow, unsaturated flow with the water table boundary, and a saturated-unsaturated flow. We also report the results of comparing such nonstationary models with Monte Carlo simulations.