A common strategy for the dimensionality reduction of nonlinear partial dif-ferential equations relies on the use of the proper orthogonal decomposition(POD) to identify a reduced subspace and the Galerkin projection for evolv-ing dynamics in this reduced space. However, advection-dominated PDEsare represented poorly by this methodology since the process of truncationdiscards important interactions between higher-order modes during time evo-lution. In this study, we demonstrate that an encoding using convolutionalautoencoders (CAEs) followed by a reduced-space time evolution by recur-rent neural networks overcomes this limitation effectively. We demonstratethat a truncated system of only two latent-space dimensions can reproduce asharp advecting shock profile for the viscous Burgers equation with very lowviscosities, and a twelve-dimensional latent space can recreate the evolutionof the inviscid shallow water equations. Additionally, the proposed frame-work is extended to a parametric reduced-order model by directly embeddingparametric information into the latent space to detect trends in system evo-lution. Our results show that these advection-dominated systems are moreamenable to low-dimensional encoding and time evolution by a CAE andrecurrent neural network combination than the POD Galerkin technique.