Computational modeling of cardiac electrophysiological signaling is of vital importance in understanding, preventing, and treating life-threatening arrhythmias. Traditionally, mathematical models incorporating physical principles have been used to study cardiac dynamical systems and can generate mechanistic insights, but their predictions are often quantitatively inaccurate due to model complexity, the lack of observability in the system, and variability within individuals and across the population. In contrast, machine-learning techniques can learn directly from training data, which in this context are time series of observed state variables, without prior knowledge of the system dynamics. The reservoir computing framework, a learning paradigm derived from recurrent neural network concepts and most commonly realized as an echo state network (ESN), offers a streamlined training process and holds promise to deliver more accurate predictions than mechanistic models. Accordingly, this research aims to develop robust ESN-based forecasting approaches for nonlinear cardiac electrodynamics, and thus presents the first application of machine-learning, and deep-learning techniques in particular, for modeling the complex electrical dynamics of cardiac cells and tissue. To accomplish this goal, we completed a set of three projects. (i) We compared the performance of available mainstream techniques for prediction with that of the baseline ESN approach along with several new ESN variants we proposed, including a physics-informed hybrid ESN. (ii) We proposed a novel integrated approach, the autoencoder echo state network (AE-ESN), that can accurately forecast the long-term future dynamics of cardiac electrical activity. This technique takes advantage of the best characteristics of both gated recurrent neural networks and ESNs by integrating a long short-term memory (LSTM) autoencoder into the ESN framework to improve reliability and robustness. (iii) We extended the long-term prediction of cardiac electrodynamics from a single cardiac cell to the tissue level, where, in addition to the temporal information, the data includes spatial dimensions and diffusive coupling. Building on the main design idea of the AE-ESN, a convolutional autoencoder was equipped with an ESN to create the Conv-ESN technique, which can process the spatiotemporal data and effectively capture the temporal dependencies between samples of data. Using these techniques, we forecast cardiac electrodynamics for a variety of datasets obtained in both in silico and in vitro experiments. We found that the proposed integrated approaches provide robust and computationally efficient techniques that can successfully predict the dynamics of electrical activity in cardiac cells and tissue with higher prediction accuracy than mainstream deep-learning approaches commonly used for predicting temporal data. On the application side, our approaches provide accurate forecasts over clinically useful time periods that could allow prediction of electrical problems with sufficient time for intervention and thus may support new types of treatments for some kinds of heart disease.