Abstract: We present new techniques for computing illumination from non-diffuse luminaires and scattering from non-diffuse surfaces. The methods are based on new closed-form expressions derived using a generalization of irradiance known as irradiance tensors. The elements of these tensors are angular moments, weighted integrals of the radiation field that are useful in simulating a variety of non-diffuse phenomena. Applications include the computation of irradiance due to directionally-varying area light sources, reflections from glossy surfaces, and transmission through glossy surfaces. The principles apply to any emission, reflection, or transmission distribution expressed as a polynomial over the unit sphere. We derive expressions for a simple but versatile subclass of these functions, called axial moments, and present complete algorithms their exact evaluation in polyhedral environments. The algorithms are demonstrated by simulating Phong-like emission and scattering effects.