This paper illustrates the themes of the title in terms of: van Kampen type
theorems for the fundamental groupoid; holonomy and monodromy groupoids; and
higher homotopy groupoids. Interaction with work of the writer is explored.Comment: 13 pages; Expansion of an invited talk given to the 7th Conference on
the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles
Ehresmann, Bedlewo 8.05.2005-15.05.2005 (Poland) Version 2: corrections of a
date and some grammar, slight referencing changes, and a small comment added
Version4. Theorem 2.2 got corrected and then uncorrected! It is now
corrected. Version5. Reference added. Various minor improvements made in
reaction to comment