We have recently passed the 75th anniversary of one of the most important results in solar and space physics: Hannes Alfvén’s discovery of Alfvén waves and the Alfvén speed. To celebrate the anniversary, this article recounts some major episodes in the history of magnetohydrodynamic (MHD) waves. Following an initially cool reception, Alfvén’s ideas were propelled into the spotlight by Fermi’s work on cosmic rays, the new mystery of coronal heating, and, as scientific perception of interplanetary space shifted dramatically and the space race started, detection of Alfvén waves in the solar wind. From then on, interest in MHD waves boomed, laying the foundations for modern remote observations of MHD waves in the Sun, coronal seismology, and some of today’s leading theories of coronal heating and solar wind acceleration. In 1970, Alfvén received the Nobel Prize for his work in MHD, including these discoveries. The article concludes with some reflection about what the history implies about the way we do science, especially the advantages and pitfalls of idealised mathematical models.