Best linear time-invariant (LTI) approximations are analysed for several interesting classes of discrete nonlinear time-invariant systems. These include nonlinear finite impulse response systems and a class of nonsmooth systems called bi-gain systems. The Frechet derivative of a smooth nonlinear system is studied as a potential good LTI model candidate. The Frechet derivative is determined for nonlinear finite memory systems and for a class of Wiener systems. Most of the concrete results are derived in an l(infinity). signal setting. Applications to linear controller design, to identification of linear models and to estimation of the size of the unmodelled dynamics are discussed.