The classical parabolic Eisenstein series is a non-holomorphic modular form of weight 0, which is associated to a cusp of a given Fuchsian group of the first kind. Recently, hyperbolic and elliptic analogs of parabolic Eisenstein series were studied by Jorgenson, Kramer and von Pippich. These are non-holomorphic modular forms of weight 0, which are associated to a geodesic or a point of the complex upper half-plane, respectively. In particular, Kronecker limit type formulas were investigated for elliptic Eisenstein series.

In the present thesis we show that hyperbolic and elliptic Eisenstein series for Hecke congruence subgroups can be realized as regularized theta lifts of non-holomorphic Poincaré series of Selberg type. More precisely, we present three different lifting results. Firstly, averaged versions of hyperbolic, parabolic and elliptic Eisenstein series are obtained as the regularized Borcherds lift of Selberg's Poincaré series in signature (2,1). Here the type of the Eisenstein series is solely determined by the sign of the index of the Poincaré series. Secondly, we realize a certain hyperbolic kernel function as a regularized Borcherds lift of a modified version of Selberg's Poincaré series in signature (2,2). Using known relations between this hyperbolic kernel function, and hyperbolic and elliptic Eisenstein series, we obtain realizations of the latter functions in terms of the mentioned Borcherds lift. Thirdly, we show that using a new Maass-Selberg type of Poincaré series as an input for the regularized Borcherds lift in signature (2,2), we obtain individual elliptic Eisenstein series.

In the final two chapters of this work we present a detailed study of the meromorphic continuation of Selberg's Poincaré series in the case of signature (2,1). Evaluating this continuation at a special harmonic point, we can express the linear term in the Laurent expansion of averaged hyperbolic, parabolic and elliptic Eisenstein series at this point in terms of certain Borcherds products. This method enables us to generalize known Kronecker limit formulas in the elliptic case to higher levels, and to establish new Kronecker limit formulas for hyperbolic Eisenstein series associated to infinite geodesics.