Facial emotion recognition (FER) is a computer vision process aimed at detecting and classifying human emotional expressions. FER systems are currently used in a vast range of applications from areas such as education, healthcare, or public safety; therefore, detection and recognition accuracies are very important. Similar to any computer vision task based on image analyses, FER solutions are also suitable for integration with artificial intelligence solutions represented by different neural network varieties, especially deep neural networks that have shown great potential in the last years due to their feature extraction capabilities and computational efficiency over large datasets. In this context, this paper reviews the latest developments in the FER area, with a focus on recent neural network models that implement specific facial image analysis algorithms to detect and recognize facial emotions. This paper’s scope is to present from historical and conceptual perspectives the evolution of the neural network architectures that proved significant results in the FER area. This paper endorses convolutional neural network (CNN)-based architectures against other neural network architectures, such as recurrent neural networks or generative adversarial networks, highlighting the key elements and performance of each architecture, and the advantages and limitations of the proposed models in the analyzed papers. Additionally, this paper presents the available datasets that are currently used for emotion recognition from facial expressions and micro-expressions. The usage of FER systems is also highlighted in various domains such as healthcare, education, security, or social IoT. Finally, open issues and future possible developments in the FER area are identified.