Weather prediction is crucial for decision-making in various social and economic sectors. The classical numerical weather prediction methods cannot incorporate the historical observations to enhance the underlying physical models, whereas the existing data-driven, deep learning-based weather prediction methods disregard either the \textbf{physics} of the weather evolution or the \textbf{topology} of the Earth's surface. In light of these disadvantages, we develop PASSAT, a novel Physics-ASSisted And Topology-informed deep learning model for weather prediction. PASSAT attributes the weather evolution to two key factors: (i) the advection process that can be characterized by the advection equation and the Navier-Stokes equation; (ii) the Earth-atmosphere interaction that is difficult to both model and calculate. PASSAT also takes the topology of the Earth's surface into consideration, other than simply treating it as a plane. Therefore, PASSAT numerically solves the advection equation and the Navier-Stokes equation on the spherical manifold, utilizes a spherical graph neural network to capture the Earth-atmosphere interaction, and generates the initial velocity fields that are critical to solving the advection equation, from the same spherical graph neural network. These building blocks constitute a deep learning-based, \textbf{physics-assisted} and \textbf{topology-informed} weather prediction mo- del. In the $5.625^\circ$-resolution ERA5 data set, PASSAT outperforms both the state-of-the-art deep learning-based weather prediction models and the operational numerical weather prediction model IFS T42.