  In analogy to classical spherical t-design points, we introduce the concept
of t-design curves on the sphere. This means that the line integral along a
t-design curve integrates polynomials of degree t exactly. For low degrees we
construct explicit examples. We also derive lower asymptotic bounds on the
lengths of t-design curves. Our main results prove the existence of
asymptotically optimal t-design curves in the Euclidean 2-sphere and the
existence of t-design curves in the d-sphere.
