'We study sufficient conditions for a decomposition system for $L^2(\R^d)$ such that it also forms a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover we obtain a norm equivalence that allows us to distinguish the membership of a distribution in these spaces by the coefficients of its expansion. Particularly we show that a nice biorthogonal wavelet system forms a unconditional basis for Triebel-Lizorkin and Besov spaces. Afterwards we apply non-linear $n$-term approximation to these bases and fully characterize the approximation spaces in terms of interpolation spaces by Jackson and Bernstein inequalities. For decomposition systems we show that a Jackson inequality can still be obtained, yielding that the interpolation space is embedded in the approximation space. Finally we give a method for construction of an unconditional basis for Triebel-Lizorkin and Besov spaces by a finite linear combination of shifts and dilates of a single function with sufficient smoothness and decay and no vanishing moments. Applying $n$-term approximation to shifts and dilates of this function we again establish a Jackson inequality. '