Introduction -- Magneto-hydrodynamics -- Challenges -- Methodologies -- Seeking the Sun’s Achilles’ Heels -- Fresh approaches -- Closing thoughts.

This book reflects on 8 decades of research on one of the longest-standing unsolved problems in modern astrophysics: why does the Sun form a hot corona? The authors give a critical overview of the field and offer suggestions on how to bridge the chasm between what we can measure, and what we can calculate. They go back to basics to explain why the problem is difficult, where we have made progress and where we have not, to help the next generation of scientists devise novel techniques to crack such a long-lasting problem. A way forward is formulated centered around refutation, using Bayesian methods to propose and to try to reject hypotheses and models, and avoiding seduction by ``confirmation bias’’. This book is aimed at physicists, students and researchers interested in understanding, learning from and solving the coronal heating problem, in an era of new dedicated facilities such as the Parker Solar Probe and the Daniel K. Inouye Solar Telescope. The book will appeal to those interested in understanding research methods and how they are changing in the modern academic environment, particular in astrophysics and Earth sciences where remote sensing is essential.