The Covid-19 pandemic has significantly spurred the development of deep learning (DL) models for the pathology automatic diagnosis based on CT scan images. However, the assumption about the generalization of the proposed models remains to be assessed and shown for concrete clinical use. In this work, we have investigated the real value of widely used public datasets for the elaboration of DL models that are dedicated to automatic diagnosis of Covid-19 using CT scans. We have collected various international public datasets from 13 countries. Different Convolutional Neural Networks (CNNs) have been trained and their performances carefully assessed. Two evaluations have been conducted: (1) an internal evaluation following a cross-validation procedure, and (2) an external evaluation on real patients coming from new and different sources. The objective is to assess the generalization capabilities considering real-world conditions: different acquisition conditions, devices and configurations. Three families from the most effective CNN models have been selected (ResNet, DenseNet and EfficientNet). These have been fine-tuned, evaluated and used within a training methodology based on transfer learning. The most effective models have been further customized in order to create new models that are dedicated to the task at hand. These models have significantly improved the diagnosis performance.