Adaptive informative path planning (AIPP) is important to many robotics applications, enabling mobile robots to efficiently collect useful data about initially unknown environments. In addition, learning-based methods are increasingly used in robotics to enhance adaptability, versatility, and robustness across diverse and complex tasks. Our survey explores research on applying robotic learning to AIPP, bridging the gap between these two research fields. We begin by providing a unified mathematical problem definition for general AIPP problems. Next, we establish two complementary taxonomies of current work from the perspectives of (i) learning algorithms and (ii) robotic applications. We explore synergies, recent trends, and highlight the benefits of learning-based methods in AIPP frameworks. Finally, we discuss key challenges and promising future directions to enable more generally applicable and robust robotic data-gathering systems through learning. We provide a comprehensive catalog of papers reviewed in our survey, including publicly available repositories, to facilitate future studies in the field.