State forecasting plays a critical role in power system monitoring, by offering system awareness even ahead of the time horizon, enhancing system observability, and providing efficient identification of the grid topology and link parameter changes. However, available approaches relying on linear estimators or single-hidden-layer feed-forward neural networks (FNNs), cannot capture long-term nonlinear dependencies in the voltage time series, and lead to suboptimal performance. To bypass these hurdles, this paper advocates deep recurrent neural networks (RNNs) for power system state forecasting. Deep RNNs capture long-term dependencies, and are easy to implement. By also leveraging the physics behind power systems, a novel architecture based on prox-linear nets (RPLN) is further developed for state forecasting based on past measurements. Simulated tests show improved performance of the proposed RNN and RPLN predictors when compared to FNN and vector autoregression based alternatives.