In this review we describe the recent progress in understanding of field theory on singular orbifolds and compare some of these results with corresponding string computations. Using orbifold trace formulae expressions for anomalies and tadpoles are obtained. For more complicated computations like the renormalization of gauge multiplets orbifold compatible field theory is developed. The gap between orbifolds and smooth compactifications is bridged by constructing explicit orbifold blowups and toric orbifold resolutions.