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1. Preliminaries. In this note we introduce a new tensor product functor in the category of complexes. We show that this tensor product is left adjoint to the Hom functor properly modified. With this tensor product we study flat complexes, pure exact sequences of complexes, pure injective complexes and give a complete description of flat pure injective complexes over a commutative noetherian ring. Also we define Gorenstein flat complexes using this new tensor product and we prove that over a commutative Gorenstein ring any complex has a Gorenstein flat cover.