A wire electrical discharge machining apparatus for discharge-machining a work (1) by relatively moving a wire electrode (5) and the work (1) by drive means (10) and by feeding a machining energy to the space between the wire electrode (5) and the work (1) by machining power feeding means (11). The wire electrical discharge machining apparatus comprises shape specifying means (15) for specifying, depending on the use of the work (1), the shape of a portion in the vicinity of an approach point (O) at which an approach path portion (4) of a relative movement path of the relative movement between the wire electrode (5) on the face to be machined of the work (1) and the work (1) and a machining shape portion (2) are connected, into a convex or concave shape to a desired degree, and adjusting means (16) for adjusting the shape of the portion in the vicinity of the approach point (O) on the basis of the specified value by the shape specifying means (15). The shape of the machined face of the work (1) in the vicinity of the approach point (O) can be adjusted to a desired shape depending on the use of the work (1).