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Deep learning techniques (in particular convolutional neural networks, CNNs) have recently emerged as a promising approach for statistical downscaling due to their ability to learn spatial features from huge spatiotemporal datasets. However, existing studies are based on complex models, applied to particular case studies and using simple validation frameworks, which makes a proper assessment of the (possible) added value offered by these techniques difficult. As a result, these models are usually seen as black boxes, generating distrust among the climate community, particularly in climate change applications. In this paper we undertake a comprehensive assessment of deep learning techniques for continental-scale statistical downscaling, building on the VALUE validation framework. In particular, different CNN models of increasing complexity are applied to downscale temperature and precipitation over Europe, comparing them with a few standard benchmark methods from VALUE (linear and generalized linear models) which have been traditionally used for this purpose. Besides analyzing the adequacy of different components and topologies, we also focus on their extrapolation capability, a critical point for their potential application in climate change studies. To do this, we use a warm test period as a surrogate for possible future climate conditions. Our results show that, while the added value of CNNs is mostly limited to the reproduction of extremes for temperature, these techniques do outperform the classic ones in the case of precipitation for most aspects considered. This overall good performance, together with the fact that they can be suitably applied to large regions (e.g., continents) without worrying about the spatial features being considered as predictors, can foster the use of statistical approaches in international initiatives such as Coordinated Regional Climate Downscaling Experiment (CORDEX). [ABSTRACT FROM AUTHOR]