This paper is part of a programme aiming at the classification of all pairs (S,G) where S is a finite linear space (or a 2-(v,k,1) design) and G is a flag-transitive automorphism group of S (flag means incident point-line pair). As shown in a forthcoming paper by Buekenhout, Delandtsheer and Doyen, the group G is necessarily either of affine or of simple type. We determine here all pairs (S,G) where G is one of the simple groups Sz(q), PSL(2,q), PSL(3,q) and PSU(3,q). The resulting linear spaces are the Desarguesian projective planes, the hermitian unitals and those derived from a projective plane P of even order provided with a complete conic C by calling points the lines of P disjoint from C and lines the points of P outside C, with the natural incidence.