  We study complex product structures on nilpotent Lie algebras, establishing
some of their main properties, and then we restrict ourselves to 6 dimensions,
obtaining the classification of 6-dimensional nilpotent Lie algebras admitting
such structures. We prove that any complex structure which forms part of a
complex product structure on a 6-dimensional nilpotent Lie algebra must be
nilpotent in the sense of Cordero-Fern\'andez-Gray-Ugarte. A study is made of
the torsion-free connection associated to the complex product structure and we
consider also the associated hypercomplex structures on the 12-dimensional
nilpotent Lie algebras obtained by complexification.
