The following preprint is available from my web page: M. Grandis Cubical cospans and higher cobordisms (Cospans in Algebraic Topology, III) Dip. Mat. Univ. Genova, Preprint 559 (2007). http://www.dima.unige.it/~grandis/wCub3.pdf http://www.dima.unige.it/~grandis/wCub3.ps Abstract. After two papers on weak cubical categories and collarable cospans, respectively, we put things together and construct a weak cubical category of cubical collared cospans of topological spaces. We also build a second structure, called a quasi cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has lax identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT's of Part II to higher cubical degree. The previous two papers of the same series are available at: http://www.tac.mta.ca/tac/volumes/18/12/18-12.dvi (ps, pdf) http://www.dima.unige.it/~grandis/wCub2.pdf Marco Grandis Dipartimento di Matematica Università di Genova Via Dodecaneso, 35 16146 Genova Italy e-mail: grandis@dima.unige.it tel: +39 010 353 6805 http://www.dima.unige.it/~grandis/