Groupoids and Crossed Objects in Algebraic Topology Ronald Brown Notes for Lectures at the Summer School on the `Foundations of Algebraic Topology', Grenoble, June 14 -July 5, 1997 (71 pages). Abstract: The notes concentrate on the background, intuition, proof and applications of the 2-dimensional Van Kampen Theorem (for the fundamental crossed module of a pair), with sketches of extensions to higher dimensions. One of the points stressed is how the extension from groups to groupoids leads to an extension from the abelian homotopy groups to non abelian higher dimensional generalisations of the fundamental group, as was sought by the topologists of the early part of this century. This links with J.H.C. Whitehead's efforts to extend combinatorial group theory to higher dimensions in terms of combinatorial homotopy theory, and which analogously motivated his simple homotopy theory. Available from http://www.bangor.ac.uk/~mas010/brownpr.html (gzipped postscript). Ronnie Brown Prof R. Brown, School of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom Tel. direct:+44 1248 382474|office: 382475 fax: +44 1248 383663 World Wide Web: home page: http://www.bangor.ac.uk/~mas010/ New article: Higher dimensional group theory Symbolic Sculpture and Mathematics: http://www.bangor.ac.uk/SculMath/ Mathematics and Knots: http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm