Numerical weather and climate simulations nowadays produce terabytes of data, and the data volume continues to increase rapidly since an increase in resolution greatly benefits the simulation of weather and climate. In practice, however, data is often available at lower resolution only, for which there are many practical reasons, such as data coarsening to meet memory constraints, limited computational resources, favoring multiple low-resolution ensemble simulations over few high-resolution simulations, as well as limits of sensing instruments in observations. In order to enable a more insightful analysis, we investigate the capabilities of neural networks to reconstruct high-resolution data from given low-resolution simulations. For this, we phrase the data reconstruction as a super-resolution problem from multiple data sources, tailored toward meteorological and climatological data. We therefore investigate supervised machine learning using multiple deep convolutional neural network architectures to test the limits of data reconstruction for various spatial and temporal resolutions, low-frequent and high-frequent input data, and the generalization to numerical and observed data. Once such downscaling networks are trained, they serve two purposes: First, legacy low-resolution simulations can be downscaled to reconstruct high-resolution detail. Second, past observations that have been taken at lower resolutions can be increased to higher resolutions, opening new analysis possibilities. For the downscaling of high-frequent fields like precipitation, we show that error-predicting networks are far less suitable than deconvolutional neural networks due to the poor learning performance. We demonstrate that deep convolutional downscaling has the potential to become a building block of modern weather and climate analysis in both research and operational forecasting, and show that the ideal choice of the network architecture depends on the type of data to predict, i.e., there is no single best architecture for all variables.