The total absolute curvature of open curves in E3 is discussed. We study the curves which attain the infimum of the total absolute curvature in the set of curves with fixed endpoints, end-directions, and length. We show that if the total absolute curvature of a sequence of curves in this set tends to the infimum, the limit curve must lie in a plane. Moreover, it is shown that the limit curve is either a subarc of a closed plane convex curve or a piecewise linear curve with at most three edges. The uniqueness of the curves minimizing the total absolute curvature is also discussed. This extends the results in [Yokohama Math. J. 48 (2000), 83–96], which deals with a similar problem for curves in E2.