Two of the main principles underlying the life cycle of an artificial intelligence (AI) module in communication networks are adaptation and monitoring. Adaptation refers to the need to adjust the operation of an AI module depending on the current conditions; while monitoring requires measures of the reliability of an AI module's decisions. Classical frequentist learning methods for the design of AI modules fall short on both counts of adaptation and monitoring, catering to one-off training and providing overconfident decisions. This paper proposes a solution to address both challenges by integrating meta-learning with Bayesian learning. As a specific use case, the problems of demodulation and equalization over a fading channel based on the availability of few pilots are studied. Meta-learning processes pilot information from multiple frames in order to extract useful shared properties of effective demodulators across frames. The resulting trained demodulators are demonstrated, via experiments, to offer better calibrated soft decisions, at the computational cost of running an ensemble of networks at run time. The capacity to quantify uncertainty in the model parameter space is further leveraged by extending Bayesian meta-learning to an active setting. In it, the designer can select in a sequential fashion channel conditions under which to generate data for meta-learning from a channel simulator. Bayesian active meta-learning is seen in experiments to significantly reduce the number of frames required to obtain efficient adaptation procedure for new frames.