The authors deal with the geometry of isometric immersions of Riemannian manifolds \(M^n(K)\) of constant sectional curvature \(K\) in analogous \((2n-1)\)-dimensional simply connected pseudo-Riemannian manifolds \(\overline{M}^{2n-1}_s(\overline{K})\) of constant sectional curvature \(\overline{K}\) with index \(s\) \((0\leq s\leq n-1)\) and \(K\neq \overline{K}\), assuming moreover the flatness of the normal bundle. The immersions are described in terms of a second order system of differential equations for a certain \(O(n-q,q)\) matrix-valued function \((q=s\) if \(K\overline{K})\) on \(M^n(K)\) which is closely related to the first and second fundamental forms; these are the so-called generalized equations (due to the fact that they involve the common sin- and sinh-Gordon equations as a very particular subcase). The Bäcklund transformation and the superposition formulae are stated with references. The intrinsic generalized equations obtained by the restriction to the first fundamental form are much simpler, however, they determine the immersions up to rigid motions of \(\overline{M}^{2n-1}_s(\overline{K})\). Some geometrical properties of immersions are found, e.g., submanifolds are foliated by \((n-1)\)-dimensional leaves of constant mean curvature in \(M(K)\), and each leaf of this foliation is itself foliated by curves of constant curvature.