  Let S be a smooth cubic surface over a field K. It is well-known that new
K-rational points may be obtained from old ones by secant and tangent
constructions. A Mordell-Weil generating set is a subset B of S(K) of minimal
cardinality which generates S(K) via successive secant and tangent
constructions. Let r(S,K) be the cardinality of such a Mordell-Weil generating
set. Manin posed what is known as the Mordell-Weil problem for cubic surfaces:
if K is finitely generated over its prime subfield then r(S,K) is finite. In
this paper, we prove a special case of this conjecture. Namely, if S contains
two skew lines both defined over K then r(S,K) = 1. One of the difficulties in
studying the secant and tangent process on cubic surfaces is that it does not
lead to an associative binary operation as in the case of elliptic curves. As a
partial remedy we introduce an abelian group H_S(K) associated to a cubic
surface S/K, naturally generated by the K-rational points, which retains much
information about the secant and tangent process. In particular, r(S, K) is
large as soon as the minimal number of generators of H_S(K) is large. In
situations where weak approximation holds, H_S has nice local-to-global
properties. We use these to construct a family of smooth cubic surfaces over
the rationals such that r(S,K) is unbounded in this family. This is the cubic
surface analogue of the unboundedness of ranks conjecture for elliptic curves.
