The presence of liquid water is a requirement of habitability on a planet. Possible indicators of liquid surface water on Mars include intermittent flow-like features observed on sloping terrains. These recurring slope lineae are narrow, dark markings on steep slopes that appear and incrementally lengthen during warm seasons on low-albedo surfaces. The lineae fade in cooler seasons and recur over multiple Mars years. Recurring slope lineae were initially reported to appear and lengthen at mid-latitudes in the late southern spring and summer and are more common on equator-facing slopes where and when the peak surface temperatures are higher. Here we report extensive activity of recurring slope lineae in equatorial regions of Mars, particularly in the deep canyons of Valles Marineris, from analysis of data acquired by the Mars Reconnaissance Orbiter. We observe the lineae to be most active in seasons when the slopes often face the sun. Expected peak temperatures suggest that activity may not depend solely on temperature. Although the origin of the recurring slope lineae remains an open question, our observations are consistent with intermittent flow of briny water. Such an origin suggests surprisingly abundant liquid water in some near-surface equatorial regions of Mars.