The space of complete collineations is an important and beautiful chapter of Algebraic Geometry, which dates back to the classical works of Chasles, Giambieli, Schubert, Semple and Tyrell in the 19th century and has been studied intensively ever since. By analogy with these classical spaces, in joint work with M. Kapranov, we introduce the variety of complete complexes. Its points can be seen as equivalence classes of spectral sequences of a certain kind. We prove that the set of such equivalence classes has the structure of a smooth projective variety, which provides a desingularization, with normal crossings boundary, of the Buchsbaum-Eisenbud variety of complexes, i.e., a so-called ``wonderful compactification'' of the union of its maximal strata.