Let U denote the set of all integers, and suppose that Y = {Yu ; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au ; u ∈ U} be a sequence of real numbers with Σu a u 2 = 1. Then Xu = Σw awY u –w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu ) = 0, E(Xu 2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au | is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞ −∞(F(y) − Φ(y))2dy ≦ g maxu |a u | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w 1, … wn ), (X w1 , … X wn ) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu |au | to the bandwidth of the filter a is studied.