In general, the answer is Yes.
Once a hurricane forms, it usually moves west. In the Northern Hemisphere, the storms then curve to the north. In the Southern Hemisphere, they usually curve to the south. The initial west motion is due to the general atmospheric wind pattern which produces a near constant east-to-west motion near the Equator where the storms most often form. The eventual shift to the north or south occurs because as the storms gradually migrate father from the equator, they come under the influence of the rotation of the earth which tends to steer winds (and storms) toward the poles.
Smaller, short-term variations to these general rules are not uncommon, especially with a large hurricane which tends to wobble and move erratically over short distances. But the general, long-range motion of these storms is almost always as described above.