Non-Fickian or “anomalous” transport, where the target’s spatial variance grows nonlinearly in time, describes the pollutant dynamics widely observed in heterogeneous geological media deviating significantly from that described by the classical advection dispersion equation (ADE). The ADE describes the Fickian-type of transport, with symmetric snapshots like the Gaussian distribution in space (Berkowitz et al., 2006). Non-Fickian transport can be observed at all scales. Non-Fickian transport is typically characterized by apparent (as heavy as power-law) early arrivals and late time tailing behaviors in the tracer breakthrough curves (BTCs). Non-Fickian transport is well known to be affected by medium heterogeneity. Heterogeneity can refer to variations in the distribution of geometrical properties, as well as variations in the biogeochemical properties of the medium, which cannot be mapped exhaustively at all relevant scales. Complex geometric structures and intrinsic heterogeneity in geological formations affect predictions of tracer transport and further challenge remediation analyses. Hence, efficient quantification of non-Fickian transport requires parsimonious models such as the fractional engine based physical models. In this dissertation, I first compared three types of time non-local transport models, which include the multi-rate mass transfer (MRMT) model, the continuous time random walk (CTRW) framework, and the tempered time fractional advection dispersion equation (tt-fADE) model. I then found that tt-fADE can model the rate-limited diffusion and sorption-desorption of Arsenic in soil. Additionally, non-Fickian dynamics for pollutant transport in field-scale discrete fracture networks (DFNs) were explored. Monte Carlo simulations of water flow were then conducted through field-scale DFNs to identify non-Darcian flow and non-Fickian pressure propagation. Finally, to address non-Fickian transport for reactive pollutants, I proposed a time fractional derivative model with the reaction term. Findings of this dissertation improve our understanding of the nature of water flow and pollutant transport in porous and fractured media at different scales. The correlated parameters and relationships between media properties and parameters can enhance the applicability of fractional partial differential equations that can be parameterized using the measurable media characteristics. This provides one of the most likely ways to improve the model predictability, which remained the most challenge for stochastic hydrologic models.