Much has been discovered about volcanism on Mars over the past fifty years of space exploration. Previous reviews of these discoveries have generally focused on the volcanic constructs (e.g., Olympus Mons and the other volcanoes within the Tharsis and Elysium regions), the analysis of individual lava flows, and how volcanic activity on Mars has evolved over time. Here we focus on attributes of volcanology that have received less attention and build upon characteristics of terrestrial volcanoes to pose new questions to guide future analyses of their Martian equivalents either with existing data sets or with new types of measurements that need to be made. The remarkable lack of exposed dikes at eroded ancient volcanoes attests to an internal structure that is different from terrestrial equivalents. Enigmatic aspects of the origin of the ridged plains (commonly accepted to be volcanic but with few identifiable flow fronts and only rare vents), the style(s) of volcanism during the earliest period of Martian history (the Noachian), and the possible mode(s) of formation of the Medusae Fossae Formation are considered here. Martian meteorites have been dated and are volcanic, but they cannot be correlated with specific geographic areas, or the chronology of Mars derived from the number of superimposed impact craters. Some of these questions about Martian volcanism can be addressed with existing instrumentation, but further progress will most likely rely on the acquisition of new data sets such as high-resolution gravity data, the return of samples from known localities, the flight of a synthetic aperture imaging radar, penetrators sent to the Medusae Fossae Formation, and detailed in situ field observations of selected volcanic sites.