Power outages in electrical grids can have very negative economic and societal impacts rendering fault diagnosis paramount to their secure and reliable operation. In this paper, deep neural networks are proposed for fault detection and location in low-voltage smart distribution grids. Due to its key properties, the proposed method solves some of the drawbacks of the existing literature methods, namely a method that: 1) is not limited by the grid topology; 2) is branch-independent; 3) can localize faults even with limited data; 4) is the first to accurately detect and localize high-impedance faults in the low-voltage distribution grid. The generalizability of the method derives from the non-grid specific nature of the inputs that it requires, inputs that can be obtained from any grid. To evaluate the proposed method, a real low-voltage distribution grid in Portugal is considered and the robustness of the method is tested against several disturbances including large fault resistance values (up to 1000 Ω). Based on the case study, it is shown that the proposed methodology outperforms conventional fault diagnosis methods: it detects faults with 100% accuracy, identifies faulty branches with 83.5% accuracy, and estimates the exact fault location with an average error of less than 11.8%. Finally, it is also shown that: 1) even when reducing the available measurements to the bare minimum, the accuracy of the proposed method is only decreased by 4.5%; 2) while deep neural networks usually require large amounts of data, the proposed model is accurate even for small dataset sizes.