The fundamental concept of colored nonautonomous solitons in nonlinear and dispersive nonautonomous physical systems is introduced. Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The parallels between nonlinear guided wave phenomena in optics and nonlinear guided wave phenomena in Bose condensates are clearly demonstrated by considering optical and matter wave soliton dynamics in the framework of nonautonomous evolution equations. The exact analytical solutions and numerical experiments reveal many specific features of nonautonomous solitons. Fundamental laws of the soliton adaptation to the external potentials are derived. Bound states of colored nonautonomous solitons are studied in detail and a comparison of the canonical Satsuma-Yajima breather dynamics with a nonautonomous 'agitated' breather is presented. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies