abstract: Let \(G\) be a simple algebraic group over \(\mathbb{C}\) with Lie algebra \(\mathfrak{g}\) and let \(e\) be a nilpotent element of \(\mathfrak{g}\). The aim of this lecture course will be the interplay between finite dimensional irreducible representations of the finite \(W\)-algebra \(U(\mathfrak{g},e)\) and primitive ideals of the universal enveloping algebra \(U(\mathfrak{g})\) whose associate variety coincides with the Zariski closure of the adjoint \(G\)-orbit of \(e\). Some applications to the representation theory of the modular counterpart of \(\mathfrak{g}\) will also be discussed.