In this thesis we study certain irrational \(\mathcal{W}\)-algebras and their representations. Main example that we investigate is the simple Bershadsky-Polyakov vertex algebra \(\mathcal{W}_k(=\mathcal{W}_k(sl_3, f_\theta))\), i.e. the minimal affine \(\mathcal{W}\)-algebra associated to \(sl_3\). T. Arakawa proved in [9] that \(\mathcal{W}_k\) is rational if \(k\) is a half integer greater that \(-3/2\). We study cases where \(\mathcal{W}_k\) is irrational, and classify \(\mathcal{W}_k\)-modules for certain \(k\). Main tools that we use to classify irreducible modules are Zhu's theory and formulas for singular vectors. Zhu algebra is realized as a quotient of the Smith algebra from [44]. Classification of irreducible strong modules for \(\mathcal{W}_k\) (i.e. modules with finite dimensional weight subspaces for the the operator \(L(0)\)) is hence connected to the classification of finite dimensional representations of the Smith algebra. In the case \(k=-5/3\), we show that \(\mathcal{W}_k\) is realized as a vertex subalgebra of the Weyl vertex algebra. We prove that in this case \(\mathcal{W}_k\) has exactly 6 irreducible strong modules. In the case \(k=-9/4\) , \(\mathcal{W}_k\) is an important example of a logarithmic vertex algebra. We classify strong \(\mathcal{W}_k\)-modules for \(k=-9/4\) and prove that \(\mathcal{W}_k\) has exactly 3 irreducible strong modules. In order to show that \(\mathcal{W}_k\) is irrational, we construct an uncountable family of weight modules for \(\mathcal{W}_k\) outside of category \(\mathcal{O}\). We construct a family of singular vectors which generalizes Arakawa’s formulas for singular vectors for \(k\) half integer from [9]. For \(k=-1\) and \(k=0\) we classify all modules in the category \(\mathcal{O}\). In the case \(k=0\), we give an explicit realization of the vertex algebra \(\mathcal{W}_k\) and its modules as certain irrational subalgebras of lattice VOAs. We also study the algebra \(\mathcal{W}_k\) when \(k\) is an integer greater than -1.