Provided are a method and apparatus for driving a liquid crystal display. The apparatus includes: a moving image detector which reads an input signal frame by frame and compares gray-scale data of a previous frame to gray-scale data of the current frame to detect a moving pattern; a gray-scale difference calculator which calculates a gray-scale difference in the detected pattern to discriminate the boundary of the pattern from the inside of the pattern; and an output processor that generates an over-driving voltage for over-driving pixels corresponding to the inside of the pattern and applies the over-driving voltage to pixels of liquid crystal.