The shape evolution and migration of <100> and <111> tilt grain boundaries with rotation angles θ in the range between 6° and 24° were investigated in situ in a scanning electron microscope at elevated temperatures. The results revealed that boundaries with misorientation θ<15° did not attain a continuously curved shape in the entire temperature range up to the melting point and, thus, did not move under a capillary driving force. Instead, they remained straight or formed several facets which were inclined to the initial boundary orientation. Molecular statics simulations suggest that the observed behavior of low-angle boundaries is due to the anisotropy of grain boundary energy with respect to boundary inclination. This anisotropy diminishes with increasing misorientation angle, and high-angle boundaries assume a continuously curved shape and move steadily under the curvature driving force.