The thesis addresses the interplay between topological dynamics, groupoids and C*- algebras. Ever since the inception of Cuntz–Krieger algebras (and later graph C*- algebras), symbolic dynamical systems have been exploited to exhibit new and interesting examples of operator algebras. Via a groupoid reconstruction theory of Kumjian and Renault (and later refined by many authors), we can now trace finer structures of the C*-algebras back to properties of the dynamical systems, and structure-preserving ∗- isomorphisms between C*-algebras back to conjugacies, orbit equivalences or flow equivalences of the dynamical systems. The first part of the thesis contains a review of the literature on this question specifically for shift spaces, while the second part contains the original contributions of the thesis. Papers A and B (joint with Toke Meier Carlsen) concern orbit equivalences and flow equivalences between shift spaces, while paper C characterizes diagonal-preserving and gauge-intertwining ∗-isomorphisms of graph C*-algebras in terms of moves on the graphs. The paper D (joint with Eduardo Scarparo) studies the topological full group of groupoids and gives conditions for these groups to be C*-simple-