Topological Methods for Ordinary Differential Equations : Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme Italy June 24–July 2 1991 The volume contains the texts of four courses given by the authors at a summer school that sought to present the state of the art in the growing field of topological methods in the theory of o.d.e. (in finite and infinitedimension) and to provide a forum for discussion of the wide variety of mathematical tools which are involved. The topics covered range from the extensions of the Lefschetz fixed point and the fixed point index on ANRs to the theory of parity of one parameter families of Fredholm operators and from the theory of coincidence degree for mappings on Banach spaces to homotopy methods for continuation principles. CONTENTS: P. Fitzpatrick: The parity as an invariant for detecting bifurcation of the zeroes of one parameter families of nonlinear Fredholm maps. M. Martelli: Continuation principles and boundary value problems. J. Mawhin: Topological degree and boundary value problems for nonlinear differential equations. R.D. Nussbaum: The fixed point index and fixed point theorems