We present a new, more nuanced understanding of non-linear effects in inverse Compton sources. Contrary to what has been heretofore understood, deleterious non-linear effects can arise even at low laser intensities, a regime previously viewed as linear. After laying out a comprehensive survey of all non-linear phenomena which degrade the effectiveness of inverse Compton sources, we discuss two powerful techniques designed to remove these non-linearities. Starting with the known technique of longitudinal chirping of the laser pulse, which we developed earlier to mitigate adverse non-linear effects in the high laser field regime, we have discovered that the simple stretching of the laser pulse, while keeping the energy constant, can significantly increase the spectral density of the scattered radiation in many operating regimes. Our numerical simulations show that combining these two techniques removes detrimental non-linearities and improves the performance of inverse Compton sources over an order of magnitude. PACS numbers: 29.20.Ej, 29.25.Bx, 29.27.Bd, 07.85.Fv X-rays enable scientists to see the internal structure of materials on all length scales from the macroscopic down to the positions of individual atoms. This capability has had profound impact on science, technology, and on the world economy. It is impossible to overstate this impact, from Nobel Prize winning science to the everyday dental x-ray. The science and technology community agrees that future advances in many areas depend on understanding structure/function relationships at the nano-scale where new properties emerge, and controlling the fabrication of complex materials at that scale to achieve transformative physical, chemical, and biological functionality.The sources of x-ray radiation relying upon Compton scattering [1, 2] possess a notable advantage over the traditional bremsstrahlung sources-the narrow-band nature of the radiation emerging from them. This motivated creation of designated facilities featuring inverse Compton sources (ICS) [3], which have been applied to x-ray structure determination [4], dark-field imaging [5,6], phase contrast imaging [5], and computed tomography [7] .As a significantly more affordable alternative to large facility sources, inverse Compton sources (ICS) of x-rays may allow a multitude of studies not easily accomplished at the large facilities. These potentially groundbreaking studies in medicine, pharmaceutical industry, chemistry, material science, homeland security and many other fields of human endeavor, hold a promise of fundamentally improving our lives. However, in order to bring about this new era, we first need to develop a deeper understanding of physical processes in ICS.In this letter, we map out the regimes of operations for ICS and describe the non-linear effects which plague each of these regimes. We then present two precise techniques which combine to largely remove these non-linearities, thereby substantially improving the peak spectral density of the scattered radiation emerging from these...