Bringing the success of modern machine learning (ML) techniques to mobile devices can enablemany new services and businesses, but also poses significant technical and research challenges. Twofactors that are critical for the success of ML algorithms are massive amounts of data and process-ing power, both of which are plentiful, yet highly distributed at the network edge. Moreover, edgedevices are connected through bandwidth- and power-limited wireless links that suffer from noise,time-variations, and interference. Information and coding theory have laid the foundations of reliableand efficient communications in the presence of channel imperfections, whose application in modernwireless networks have been a tremendous success. However, there is a clear disconnect between thecurrent coding and communication schemes, and the ML algorithms deployed at the network edge. Inthis paper, we challenge the current approach that treats these problems separately, and argue for a jointcommunication and learning paradigm for both the training and inference stages of edge learning.