We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for $n$-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over- and under-crossings. We demonstrate the power of our approach by applying it to $��^4$-theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised.

31 pages, LaTeX with AMS and XY-Pic macros; final version to appear in Commun. Math. Phys