Recently the authors and D. E. Blair introduced in [Isr. J. Math. 141, 157–183 (2004; Zbl 1064.53026)] the notion of a generalized Sasakian space form. In the present paper the submanifolds of such a space form are studied. It is focused on almost semi-invariant submanifolds, which were initiated by M. M. Tripathi and K. D. Singh [Demonstr. Math. 29, No. 2, 413–426 (1996; Zbl 0896.53041)], followed by M. M. Tripathi and I. Mihai [Note Mat. 20(2000–2001), No. 2, 135–164 (2001; Zbl 1097.53037)], and generalize invariant, anti-invariant, and slant submanifolds. Sectional curvatures, Ricci tensor and scalar curvature are also studied. The paper finishes with some results about totally umbilical submanifolds.