This thesis investigates the dimensional crossover in low-dimensional interacting systems. To this end, we employ a field theory approach called bosonization to study a toy model consisting of coupled one-dimensional chains. As we tune the coupling between the chains, the system develops coherence that would not be present in a purely one-dimensional setup. These features are encoded in the one-body correlation function computed with a variational approach, but also in the transverse superfluid fraction. We find the relevant energy scales and validate our approach with accurate numerical simulations (Quantum Monte Carlo) to provide a comprehensive understanding of the underlying physics. The results are then successfully tested with ultracold atom setups which allow for tunable interaction strength while continuously monitoring the system undergoing the dimensional crossover.