Damage diagnosis of thin-walled structures has been successfully performed through methods based on tomography and machine learning-driven methods. According to traditional approaches, diagnostic signals are excited and sensed on the structure through a permanently installed network of sensors and are processed to obtain information about the damage. Good performance characterizes methods that process Lamb waves, which are described by long propagation distances and high sensitivity to anomalies. Most of the methods require extracting damage-sensitive features from the diagnostic signals to drive the damage diagnosis task. However, this process can lead to loss of information, and the choice of the specific feature to extract may introduce biases that hamper damage diagnosis. Furthermore, traditional approaches do not perform well when composites are considered, due to the anisotropy, inhomogeneity, and complex damage mechanisms shown by this type of material. To boost the performance of methods for damage diagnosis of composite plates, this work proposes a convolutional neural network (CNN)-based algorithm that localizes damage by processing Lamb waves. Different from other methods, the proposed method does not require extracting features from the acquired signals and allows localizing damage through the regression approach. The method was tested against experimental observations of Lamb waves propagating in two composite panels and in a hybrid panel, and the performance of two different sensor arrays was investigated. The pseudo-damage approach was used to generate large enough datasets for training the CNNs, and the performance of the framework was evaluated by localizing pseudo-damage and real damage determined by low-velocity impacts. The CNN-driven method accurately localized damage in all the considered scenarios, and it also outperformed traditional damage indices-based approaches, such as the reconstruction algorithm for probabilistic inspection of defects.