The state-of-the-art of insect flight research using advanced computational fluid dynamics techniques on supercomputers is reviewed, focusing mostly on the work of the present authors. We present a brief historical overview, discuss numerical challenges and introduce the governing model equations. Two open source codes, one based on Fourier, the other based on wavelet representation, are succinctly presented and a mass-spring flexible wing model is described. Various illustrations of numerical simulations of flapping insects at low, intermediate and high Reynolds numbers are presented. The role of flexible wings, data-driven modeling and fluid-structure interaction issues are likewise discussed.