CiE 2009: Mathematical Theory and Computational Practice Heidelberg, Germany, July 19-24, 2009 After several years of research activity, the informal cooperation "C- putability in Europe" decided to take a more formal status at their meeting in Athens in June 2008: the Association for Computability in Europe was founded to promote the development, particularly in Europe, of computability-related science, ranging over mathematics, computer science, and applications in va- ous natural and engineering sciences such as physics and biology, including the promotion of the study of philosophy and history of computing as it relates to questionsofcomputability. As mentioned, this associationbuilds ontheinformal network of European scientists working on computability theory that had been supporting the conference series CiE-CS overthe years, and nowbecame its new home. The aims of the conference series remain unchanged: to advance our t- oretical understanding of what can and cannot be computed, by any means of computation. Its scienti?c vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs, and - chines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory or relativity. Computations may be very general, depending on the foundations of set theory; or very speci?c, using the combinatorics of ?nite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations.

These proceedings contain research and survey papers from many subfields of recursion theory, with emphasis on degree theory, in particular the development of frameworks for current techniques in this field. Other topics covered include computational complexity theory, generalized recursion theory, proof theoretic questions in recursion theory, and recursive mathematics.