The work can be considered as an attempt to generalize the results of \textit{F. Lalonde} and \textit{J.-C. Sikorav} [Comment. Math. Helv. 66, No. 1, 18--33 (1991; Zbl 0759.53022)] who invented Lagrange surgery for surfaces. The following aspects of the existence question for Lagrange embeddings are discussed: what manifolds admit a Lagrange embedding into \(\mathbb{C}^ n\), given a symplectic manifold; what middle-dimensional homology classes can be represented by a Lagrange embedding? New characteristic classes of a generic Lagrange immersion \(f: L\to \mathbb{C}^ n\) are constructed by using a Lagrange surgery. New constructions of embedded Lagrange submanifolds of \(\mathbb{C}^{2k}\) \((k>1)\) are presented which are diffeomorphic to \(S^{2k-1}\times S^ 1\) and belong to different connected components of the space of Lagrange immersions.