This volume reflects the growing collaboration between mathematicians and theoretical physicists to treat the foundations of quantum field theory using the mathematical tools of q-deformed algebras and noncommutative differential geometry. A particular challenge is posed by gravity, which probably necessitates extension of these methods to geometries with minimum length and therefore quantization of space. This volume builds on the lectures and talks that have been given at a recent meeting on "Quantum Field Theory and Noncommutative Geometry." A considerable effort has been invested in making the contributions accessible to a wider community of readers - so this volume will not only benefit researchers in the field but also postgraduate students and scientists from related areas wishing to become better acquainted with this field.