a b s t r a c tSuppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m = 4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].An automorphism φ of a group G is said to be almost fixed-point-free (a phrase we shorten to afpf) if its fixed-point set C G (φ) is finite. In [3] Endimioni proves that a polycyclic-by-finite group with an afpf automorphism of prime order is nilpotent-by-finite. Here we present a somewhat shorter alternative proof of this that also delivers the same conclusion for finite extensions of torsion-free soluble minimax groups and also for soluble-by-finite linear groups over finite extension fields of the rational numbers Q. Trivially polycyclic groups are (torsion-free)-by-finite, soluble, minimax and of finite rank and, less trivially (e.g. see [7], 1.2), (torsion-free)-by-finite soluble groups of finite rank have faithful finite-dimensional representations over the rationals Q. Neither reverse implication holds.If G is a finite soluble group with a fixed-point-free automorphism of prime order p, then G is nilpotent of class bounded by a function of p only-see Higman [4]; the least such function we denote here by h(p). (By a famous theorem of J.G. Thompson the solubility restriction is unnecessary, but we only require the soluble case.) Endimioni actually proves that if G is polycyclic-by-finite with an afpf automorphism of prime order p, then G has a nilpotent normal subgroup of finite index and class at most h(p). This also applies to our minimax, finite rank and linear cases. Below, Theorem 1 is our basic result and its proof delivers our proof of Endimioni's theorem. Theorem 2 we derive from Theorem 1 and Corollary 1 is immediate from Theorem 2 and [7], 1.2.