In this thesis we study certain affine vertex algebras beyond admissible levels. Our primary focus is the simple affine vertex algebra \(L_{-5/2} (sl(4))\). We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra \(V^{-5/2} (sl(4))\)and show that it generates the maximal ideal in \(V^{-5/2} (sl(4))\). We classify irreducible \(L_{-5/2} (sl(4))\)–modules in the category \(\mathscr{O}\), and determine the fusion rules between irreducible modules in the category of ordinary modules \(KL_{-5/2}\). We also prove that \(KL_{-5/2}\) is a semi-simple tensor category. In our proofs we use an explicit formula for singular vectors and Zhu’s theory. We also use properties of conformal embedding \(gl(4) \hookrightarrow sl(5)\) at level \(k = -5/2\) [6] and the theory of affine \(\mathcal{W}\) –algebras. We show that \(k = -5/2\) is a collapsing level with respect to the subregular nilpotent element \(f_{subreg}\) and prove certain results on vanishing and non-vanishing of the quantum Hamiltonian reduction functor \(H_{f_{subreg}}\). In the second part of this thesis we give a new approach to the classification of irreducible modules in the category \(\mathscr{O}\) for simple affine vertex algebras beyond admissible levels. We introduce the notion of almost admissible and principal almost admissible weights and show how they can be used to describe irreducible modules in the category \(\mathscr{O}\) for vertex algebra \(L_{-5/2}(sl(4))\). We also show that such an approach can be applied to vertex algebras \(L_{-1}(sl(n))\), for \(n \geq 3\), whose classification of irreducible modules in the category \(\mathscr{O}\) was determined in [12]. For the above algebras, we construct a large family of modules to which the Kac-Wakimoto character formula applies. In the end, we study the vertex algebra \(L_{-7/2}(sl(6))\) and give the conjecture for the higher-rank cases.