We will introduce adaptive Fourier decomposition (AFD) and mono-component theory. The topics were motivated by positive-frequency-decomposition of signals. It adaptively selects the parameters defining a orthogonal rational system (TM system) according to the given signal to be decomposed (the identically zero-selections gives rise to the Fourier system). The related issues include mono-components in relation to instantaneous frequency of functions, variations of AFD (unwinding ADF, cyclic AFD), AFD in higher dimensions with the Clifford and several complex variables settings, Bedrosian and non-Bedrosian type mono-components, theory and applications of Beurling’s backward shift invariant spaces in band preserving, phase retrieving and in solving the Bedrodian equation, applications of analytic phase derivative and Hardy-Sobolev derivatives in proving extra-strong uncertainty principle of the Heisenberg type, and pre-orthogonal greedy algorithm in a Hilbert space with a dictionary.