The following preprint is available: M. Grandis, Higher fundamental groupoids for spaces, Dip. Mat. Univ. Genova, Preprint 447 (Jan 2002). (17 p.) ftp://www.dima.unige.it/Home/grandis/public/HGpd.ps Abstract. Fundamental n-groupoids for a topological space are introduced, by techniques based on Moore paths, similar to those used in a previous paper for symmetric simplicial sets (M. Grandis, Higher fundamental functors for simplicial sets, Cahiers Topologie Geom. Differentielle Categ. 42 (2001), 101-136). Also the 'directed case' is treated, based on a structure recently introduced: a 'directed topological space', where privileged directions are assigned and paths need not be reversible (M. Grandis, Directed homotopy theory, I. The fundamental category, Dip. Mat. Univ. Genova, Preprint 443). Such objects are provided here with fundamental n-categories, as it was done for ordinary simplicial sets in the first cited paper. We end by comparing the present structures with the previous ones, via a geometric realisation of symmetric and ordinary simplicial sets, as spaces and directed spaces, respectively. All this essentially agrees also with the classical treatment of Kan complexes as non-directed structures. _____ Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/ ftp://www.dima.unige.it/Home/grandis/public/