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 ANR's, 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.

The past forty years have been the stage for the maturation of mathematical biolo as a scientific field. The foundations laid by the pioneers of the field during the first half of this century have been combined with advances in ap plied mathematics and the computational sciences to create a vibrant area of scientific research with established research journals, professional societies, deep subspecialty areas, and graduate education programs. Mathematical biology is by its very nature cross-disciplinary, and research papers appear in mathemat ics, biology and other scientific journals, as well as in the specialty journals devoted to mathematical and theoretical biology. Multiple author papers are common, and so are collaborations between individuals who have academic bases in different traditional departments. Those who seek to keep abreast of current trends and problems need to interact with research workers from a much broader spectrum of fields than is common in the traditional mono-culture disciplines. Consequently, it is beneficial to have occasions which bring together significant numbers of workers in this field in a forum that encourages the exchange of ideas and which leads to a timely publication of the work that is presented. Such an occasion occurred during January 13 to 16, 1990 when almost two hun dred research workers participated in an international conference on Differential Equations and Applications to Biology and Population Dynamics which was held in Claremont."

The meeting explored current directions of research in delay differential equations and related dynamical systems and celebrated the contributions of Kenneth Cooke to this field on the occasion of his 65th birthday. The volume contains three survey papers reviewing three areas of current research and seventeen research contributions. The research articles deal with qualitative properties of solutions of delay differential equations and with bifurcation problems for such equations and other dynamical systems. A companion volume in the biomathematics series (LN in Biomathematics, Vol. 22) contains contributions on recent trends in population and mathematical biology.