  In continuum physics, there are important topological aspects like
instantons, theta-terms and the axial anomaly. Conventional lattice
discretizations often have difficulties in treating one or the other of these
aspects. In this paper, we develop discrete quantum field theories on fuzzy
manifolds using noncommutative geometry. Basing ourselves on previous
treatments of instantons and chiral fermions (without fermion doubling) on
fuzzy spaces and especially fuzzy spheres, we present discrete representations
of theta-terms and topological susceptibility for gauge theories and derive
axial anomaly on the fuzzy sphere. Our gauge field action for four dimensions
is bounded by the modulus of the instanton number as in the continuum.
