Let \((M,\xi:={Ker}(\alpha))\) be a contact \((2n+1)\)-manifold, that is \(\alpha\) is a \(1\)-form on \(M\) such that \(\alpha\wedge{(d\alpha)}^n\neq 0\). The standard contact \((2n+1)\)-space is \((\mathbb{R}^{2n+1},\xi:={Ker}(\alpha))\), where \(\alpha\) is the (contact) \(1\)-form \(\alpha:=dy-\sum^n_{j=1}x^{2j}dx^{2j-1}\) in Euclidean coordinates \((x^1,x^2,\dots,x^{2n-1},x^{2n},y)\). An immersion of an \(n\)-manifold into \((M,\xi)\) is called Legendrian if it is everywhere tangent to the hyperplane field \(\xi\). The image of a Legendrian embedding is called a Legendrian submanifold. One can show – via the \(h\)-principle for Legendrian immersions (see [M. Gromov, Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 9. (Berlin) etc.: Springer-Verlag. (1986; Zbl 0651.53001)]) – that Legendrian submanifolds of standard contact \((2n+1)\)-space exist in abundance. The authors study the following question: When are two Legendrian submanifolds of the standard contact \((2n+1)\)-space isotopic through Legendrian submanifolds? For \(n=1\), this question has been extensively studied [see Y. Chekanov, Invent. Math. 150, No.3, 441-483 (2002; Zbl 1029.57011); Y. Eliashberg and M. Fraser, Geometry, topology, and dynamics. Proceedings of the CRM workshop, Montreal, Canada, June 26–30, 1995. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 15, 17–51 (1998; Zbl 0907.53021); J. B. Etnyre and K. Honda, J. Symplectic Geom. 1, No. 1, 63–120 (2001; Zbl 1037.57021) and Adv. Math. 179, No. 1, 59–74 (2003; Zbl 1047.57006); J. B. Etnyre, L. L. Ng and J. M. Sabloff, J. Symplectic Geom. 1, No. 2, 321–367 (2002; Zbl 1024.57014)]. When \(n>1\), the authors define two classical invariants of an oriented Legendrian submanifold given by an embedding \(f:L\to \mathbb{R}^{2n+1}\), namely the Thurston-Bennequin invariant [following S. Tabachnikov, Russ. Math. Surv. 43, No. 3, 225–226 (1988); translation from Usp. Mat. Nauk 43, No. 3(261), 193–194 (1988; Zbl 0667.57017)] and the rotation class, and they prove the following theorem: For any \(n>1\), there is an infinite family of Legendrian embeddings of the \(n\)-sphere into \((\mathbb{R}^{2n+1},\xi)\) that are pairwise not Legendrian isotopic even though they have the same classical invariants. Concerning this result, one should note that there is an important distinction from the three dimensional case, see [V. Colin, E. Giroux and K. Honda, Topology and geometry of manifolds. Proceedings of the 2001 Georgia topology conference, University of Georgia, Athens, GA, USA, May 21–June 2, 2001. Providence, RI: American Mathematical Society (AMS). Proc. Symp. Pure Math. 71, 109–120 (2003; Zbl 1052.57036)], for details. A similar result for Legendrian surfaces and \(n\)-tori by explicitly constructing such infinite families is proven. When \(n\) is even, these are the first known examples of non-Legendrian isotopic, Legendrian submanifolds of the standard contact \((2n+1)\)-space. The authors’ examples are based on two constructions: stabilization and front spinning. To show that Legendrian submanifolds are not Legendrian isotopic, the contact homology of a Legendrian submanifold in the standard contact \((2n+1)\)-space is computed. This is done because Legendrian submanifolds with different contact homologies could not be isotopic (see the authors, Legendrian submanifolds in \(\mathbb{R}^{2n+1}\) and contact homology, Part 2, preprint, 2002, ArXiv: math. SG/0210124, for details).