A Sasakian space form is defined as a complete simply-connected Sasakian manifold of constant \(\phi\)-sectional curvature and as such is one of the model spaces described by \textit{S. Tanno} [Tôhoku Math. J., II. Ser. 21, 501-507 (1969; Zbl 0188.268)]. A Sasakian space form of constant \(\phi\)- sectional curvature c and dimension \(2m+1\) is denoted by \(M^{2m+1}(c).\) The authors prove the following theorems: 1. A compact Sasakian submanifold of \(M^{2m+1}(c)\) with \(c>-3\) and \(\phi\)-sectional curvature greater than (c-3)/2 is totally geodesic. 2. A complete Sasakian submanifold \(N^{2n+1}\) (n\(\geq 2)\) of \(M^{2m+1}(c)\), \(c>-3\), with sectional curvature greater than \((c+3)/8\) is totally geodesic. 3. Let \(N^{m+1}\) (m\(\geq 2)\) be a compact minimal anti-invariant submanifold of \(M^{2m+1}(c)\), \(c>-3\), such that N is tangent to the structure vector field \(\xi\) of M. If the sectional curvature of N of plane sections normal to \(\xi\) is positive, then N is c-totally geodesic and hence locally a Riemannian product \(\Sigma^ m\times \Sigma^ 1\) where \(\Sigma^ m\) is a totally geodesic anti-invariant submanifold of M(c) and \(\Sigma^ 1\) is generated by \(\xi\). 4. Let \(N^ m\) be a compact minimal submanifold of \(M^{2m+1}(c)\), \(c>-3\), such that N is an integral submanifold of the contact distribution (subbundle). If N has positive curvature, then N is totally geodesic. The proofs of the first three theorems make use of the regularity of the model space which gives that space a fibering over a Kaehler manifold; the results then follow from the corresponding results for Kaehler manifolds. The proof of the fourth theorem uses a function defined on the unit tangent bundle of N in terms of the second fundamental form of N in M(c) [cf. \textit{A. Ros}, Proc. Am. Math. Soc. 93, 329-331 (1985; Zbl 0561.53055)].