In lock-up capacity control apparatus and method for a torque converter, the torque converter is caused to have a converter state in which a relative revolution between input and output elements of the torque converter is free of limitation in a case where a lock-up capacity (L/Uprso) during a no-load state is equal to or larger than a shock determining lock-up capacity (.alpha.), during an acceleration slip lock up in which the lock-up capacity of the torque converter is augmented by means of a time series control which accords with a load state of an engine, and the torque converter is caused to be oriented toward a lock-up state in which the relative revolution is zeroed by continuing an augmentation of the lock-up capacity by means of the time series control in a case where the lock-up capacity during the no-load state is smaller than the shock determining lock-up capacity.