TheHamiltonian systemwhiteStochastic averagingnoiseStochastic averaging method of quasi-Hamiltonian system does not exist in reality due to its infinite energy, and it is merely an idealized mathematical model. Real noises are always colored noisesColored noise. One class of models for colored noisesColored noise is generated from Gaussian white noiseGaussian white noiseusing linear orNonlinear filter nonlinear filters (see Sects. 2.6.1 and 2.6.2  in vol. 1). A noise generated from a linear filterLinear filter is often referred to as a rational noiseRational noise because its power spectral density is a rational function of frequency. Fractional Gaussian noisesFractional Gaussian noise are another type of colored noisesColored noise generated by Gaussian white noiseGaussian white noise through a special filter. Colored noise can be wideband or narrowband. A colored noiseColored noise can also be wideband in one frequency bandFrequency band and narrowband in the others, for instance, fractional Gaussian noisesFractional Gaussian noise (see Sect. 2.5.3 in vol. 1). Narrowband noises can also be synthesized by combining harmonic and white noise or wideband noise, or by randomizing harmonic functions (see Sect. 2.6.3 in vol. 1). This chapter presents the stochastic averagingStochastic averaging methods of quasi-integrable Hamiltonian systemsHamiltonian system excited by four classes ofColored noise colored noises: stationary wideband noiseStationary wideband noise, fractional Gaussian noiseFractional Gaussian noise in the wideband frequency domain, combination of harmonic and wideband noise, andNarrowband randomized harmonic noise narrowband randomized harmonic noiseHarmonic noise.