Fault detection is crucial in smart grid control and monitoring operations. The use of smart meters leads to appearance of a large amount of digital data whose conventional and chronological techniques are not efficient enough for processing and decision-making. In this paper, a novel data analysis model based on deep learning and neuro-fuzzy algorithm is proposed for detection and classification of faults in a smart grid. First, the Long Short Term Memory (LSTM) based deep learning model is applied for training the data samples extracted from the smart meters. Then, the Adaptive Neuro Fuzzy Inference System (ANFIS) is implemented for fault detection and classification from the trained data. With this intelligent method proposed, single-phase, two-phase and three-phase faults can be identified using a restricted amount of data. To verify the effectiveness of our methodology, an intelligent model of the IEEE 13-node network is used. The results indicate that the combined ANFIS-LSTM deep learning model outperforms existing machine learning methods in the literature in terms of accuracy for fault detection and classification.