Computational cardiology relies on the solution of complex partial differential equation models to simulate cardiac electrophysiology from non-invasive measurements. However, high-fidelity simulations on realistic geometries are computationally demanding, and clinical applications require fast, interpretable, and task-oriented outputs. In this talk, we present recent advances in scientific machine learning for both forward and inverse cardiac problems, with a focus on operator learning and neural surrogate modeling. First, we introduce operator learning approaches, namely Fourier Neural Operators (FNOs) and Kernel Operator Learning (KOL), to directly learn the mapping from activation regions in the physical domain to cardiac activation and repolarization times. These surrogate operators are trained on synthetic 2D and 3D domains as well as on a physiologically realistic left ventricle geometry. While the learned mapping to activation times is consistent with the Eikonal model, the repolarization-time operator has no direct PDE counterpart, highlighting the flexibility of data-driven operator learning. Second, we address cardiac inverse problems, targeting the reconstruction of ischemic regions and stimulus locations from pseudo-ECG signals. Here, an architecture inspired by Latent Dynamics Networks (LDNets) act as fast neural surrogates of psudoECG derived from the monodomain model enabling efficient forward evaluations within an inverse learning framework on both 2D and 3D ventricular geometries. Overall, these results demonstrate that machine-learning-based surrogate and operator models can substantially accelerate cardiac simulations and inverse reconstructions, offering promising tools for clinically relevant applications.