Summary: Machine learning and logical reasoning have been the two foundational pillars of Artificial Intelligence (AI) since its inception, and yet, until recently the interactions between these two fields have been relatively limited. Despite their individual success and largely independent development, there are new problems on the horizon that seem solvable only via a combination of ideas from these two fields of AI. These problems can be broadly characterized as follows: how can learning be used to make logical reasoning and synthesis/verification engines more efficient and powerful, and in the reverse direction, how can we use reasoning to improve the accuracy, generalizability, and trustworthiness of learning. In this perspective paper, we address the above-mentioned questions with an emphasis on certain paradigmatic trends at the intersection of learning and reasoning. Our intent here is not to be a comprehensive survey of all the ways in which learning and reasoning have been combined in the past. Rather we focus on certain recent paradigms where corrective feedback loops between learning and reasoning seem to play a particularly important role. Specifically, we observe the following three trends: first, the use of learning techniques (especially, reinforcement learning) in sequencing, selecting, and initializing proof rules in solvers/provers; second, combinations of inductive learning and deductive reasoning in the context of program synthesis and verification; and third, the use of solver layers in providing corrective feedback to machine learning models in order to help improve their accuracy, generalizability, and robustness with respect to partial specifications or domain knowledge. We believe that these paradigms are likely to have significant and dramatic impact on AI and its applications for a long time to come.