 The paper discusses how reactive and dissipative non-linearities affect the intrinsic response of superconducting thin film resonators, and explains how most complex phenomena can be described by a model in which the underlying resonance is a single-pole Lorentzian with changing center frequency and quality factor as external parameters are varied. The paper provides detailed formulations for each case and explains how to extract important parameters from distorted swept frequency measurements. This article was authored by C. N. Thomas, S. Wavington, Z. Sun, and others.