Many systems, classical or quantum, closed or open, exhibit universal properties. In the context of quantum matter, Bose-Einstein condensation emerges as an emblematic physical phenomenon, where the lowest quantum energy state is macroscopically occupied as an external control parameter is tuned. In this thesis, we focus on a type of driven-dissipative condensate formed by exciton-polaritons. These quasi-particles are created through the continuous optical driving of a semiconductor microcavity, where injected photons strongly interact with electron-hole pairs, leading to a condensation transition above a certain threshold of the driving amplitude. We investigate the emergence of key features of low-dimensional quasi Bose-Einstein condensates (BECs) in exciton-polariton systems, in both one and two dimensions. An important and extensively studied feature of weakly interacting BECs lies in their ground-state energy, particularly in how it is affected by quantum fluctuations. Such fluctuations induce a small energy shift in the condensate spectrum, which brings a correction to the predictions of mean-field descriptions. We predict an analogous effect in exciton-polariton condensates, induced by stochastic fluctuations arising from repeated gain and loss events in the semiconductor microcavity. In the absence of topological defects, we compute the stochastic corrections to the condensate blueshift within the Bogoliubov approximation. These predictions are then compared with numerical simulations of a one-dimensional polariton condensate. In this context, we emphasize that blueshift corrections can be interpreted as the asymptotic velocity of a stochastically growing interface. This interpretation arises from mapping the effective phase dynamics onto a Kardar-Parisi-Zhang (KPZ) equation, which describes the universal properties of kinetically roughening surfaces. A second prominent characteristic of Bose-Einstein condensates lies in their coherence properties. In two dimensions and at low temperatures, they exhibit a quasi-ordered phase characterized by an algebraic decay of the first-order correlation function. Above a critical temperature, as described by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism, vortex–antivortex pairs unbind within the condensate, leading to a disordered phase with exponentially decaying correlations. In recent years, a similar mechanism has been proposed for exciton-polaritons. It has been argued to compete with the emerging Kardar-Parisi-Zhang (KPZ) physics, eventually hindering the latter. While several studies have suggested that a KPZ phase could still be accessible in experiments, a unified understanding of these competing scenarios under realistic conditions has remained elusive. Through extensive numerical simulations, we demonstrate that the effective nonlinearity of the condensate phase dynamics can be finely tuned over a wide range by varying the exciton-polariton interaction strength. This tuning enables the exploration of three main universal regimes with parameters compatible with current experimental capabilities: the weakly nonlinear Edwards-Wilkinson (EW) regime, where the phase fluctuations dominate, but the phase profile does not become rough, the strongly non-linear Kardar-Parisi-Zhang regime, where the condensate phase fluctuations grow in a superdiffusive manner leading to roughening of the phase profile, and a vortex-dominated phase emerging at stronger interactions, where both density and phase dynamics play significant roles. The signature of each of these phases is highlighted by the decay of the condensate first order correlation function, its finite size effects, as well as its fluctuation statistics.