In this thesis we have investigated representation theory of two important generalizations of the Hesienberg-Virasoro Lie algebra \(\mathcal{H}\), which are the N = 1 super Heisenberg-Virasoro Lie algebra \(\mathcal{SH}\) and the Schrödinger-Virasoro Lie algebra \(\mathfrak{sv}\). The former provides an example of an infinite dimensional Lie super-algebra which has not been yet discovered, while the latter is also an example of an infinite dimensional Lie algebra that has been widely studied in the physical as well as mathematical literature because it is the first of an infinite series of the so called Galilean Lie conformal algebras extended by the non-relativistic spin parametre. The important innovation in this thesis is the explicit construction of an isomorphism between the universal (resp. simple) enveloping vertex superalgebra \(V^{ \mathcal{SH}}(c_L, c_{L,\alpha}, c_\alpha \neq 0)\) (resp. \(L^{ \mathcal{SH}}(c_L, c_{L,\alpha}, c_\alpha \neq 0)\)) associated to the N = 1 super Heisenberg-Virasoro Lie algebra \(\mathcal{SH}\) and the vertex algebra \(SM(1) \bigotimes V^{\mathcal{NS}}_{c_L-c_\mu}\), (resp. \(SM(1) \bigotimes L^{\mathcal{NS}}_{c_L-c_\mu}\)), where we have shown that there is a representation of the simple N = 1 Heisenberg-Virasoro vertex-algebra \(L^{ \mathcal{SH}}(\frac{3}{2}-3\mu^2, \frac{\mu}{2}c_\alpha, c_\alpha), c_\alpha \neq 0\) on \(SM(1)\). As far as the realization of this vertex algebra at level zero case \(c_\alpha=0\), this is a continuation of the papers by Dražen Adamović and Gordan Radobolja from Journal of Pure and Applied Mathematics (2015) and Communications in Contemporary Mathematics (2019), where free field realizations of the Heisenberg-Virasoro Lie algebra at level zero had been constructed. In this thesis, we have classified singular vectors on levels \(\frac{1}{2}, 1, \frac{3}{2}, 2\) and 4 and we have also given a construction of the screening operators for this algebra, which has enabled us to prove the existence of an infinite series of singular vectors for the N = 1 Heisenberg-Virasoro Lie algebra \(\mathcal{SH}\) at the levels of conformal weight \(\frac{p}{2}\); for \(p\) odd. In order to do so, we have studied realizations of the Verma modules for this algebra in the vertex algebra associated to a certain lattice as well as intertwining operators and the Schur polynomials, which led us to the explicit formulae for this singular vectors. In the case of the Schrödinger-Virasoro Lie algebra \(\mathfrak{sv}\), we have provided a rigorous treatment as well as a generalization of the result of Unterberger from his paper Nuclear Physics B (2009). Furthermore, we have proved that the vertex algebra \(V^{\mathfrak{sv}}(c_L, c_{L,\beta}, c_\beta)\) contains a Heisenberg-Virasoro subalgebra and we have found an example of a singular vector at the level three conformal weight of the universal Schrödinger-Virasorove vertex algebra for the case of general central charges.