Smart grids are able to forecast customers’ consumption patterns, i.e., their energy demand, and consequently electricity can be transmitted after taking into account the expected demand. To face today’s demand forecasting challenges, where the data generated by smart grids is huge, modern data-driven techniques need to be used. In this scenario, Deep Learning models are a good alternative to learn patterns from customer data and then forecast demand for different forecasting horizons. Among the commonly used Artificial Neural Networks, Long Short-Term Memory networks—based on Recurrent Neural Networks—are playing a prominent role. This paper provides an insight into the importance of the demand forecasting issue, and other related factors, in the context of smart grids, and collects some experiences of the use of Deep Learning techniques, for demand forecasting purposes. To have an efficient power system, a balance between supply and demand is necessary. Therefore, industry stakeholders and researchers should make a special effort in load forecasting, especially in the short term, which is critical for demand response.