Bruns studied mathematics, astronomy, and physics at the University of Berlin during 1866–1871 under Ernst Kummer and Karl Weierstrass and earned a doctoral degree with a dissertation titled De proprietate quadam functionis potentialis corporum homogeneorum ("On the properties of a certain potential function of homogeneous bodies").

Heinrich Bruns was mainly engaged in developing the theoretical side of "the shape of the Earth" (the title of one of his major works).

The fields of potential theory and the study of equilibrium shapes owe many key results to him, including Bruns formula.

For the study of astronomical refraction he developed an unusual method of calculating the vertical gradient of air temperature together with his assistant Felix Hausdorff.

The 20th century's higher geodesy (a sub-field of geodesy concerned with measuring the earth on a global scale) as practiced by Karl Ledersteger was based on theories developed by Bruns, including "Bruns' polyhedron".