Context. The Helioseismic and Magnetic Imager (HMI) provides continuum images and magnetograms with a cadence better than one per minute. It has been continuously observing the Sun 24 h a day for the past 7 yr. The trade-off between full disk observations and spatial resolution means that HMI is not adequate for analyzing the smallest-scale events in the solar atmosphere. Aims: Our aim is to develop a new method to enhance HMI data, simultaneously deconvolving and super-resolving images and magnetograms. The resulting images will mimic observations with a diffraction-limited telescope twice the diameter of HMI. Methods: Our method, which we call Enhance, is based on two deep, fully convolutional neural networks that input patches of HMI observations and output deconvolved and super-resolved data. The neural networks are trained on synthetic data obtained from simulations of the emergence of solar active regions. Results: We have obtained deconvolved and super-resolved HMI images. To solve this ill-defined problem with infinite solutions we have used a neural network approach to add prior information from the simulations. We test Enhance against Hinode data that has been degraded to a 28 cm diameter telescope showing very good consistency. The code is open source.