Synchronization is one of the fascinating emergent phenomena in complex dynamical systems comprising a large population of interacting degrees of freedom. This phenomenon abounds in nature. We study such an emergent behavior within the ambit of the celebrated Kuramoto model. A nearest-neighbor interaction in the model affects the synchronization dynamics dramatically, which we demonstrate in three different contexts. First, we explore the effect of competing interactions by including a nearest-neighbor interaction in the Kuramoto model on a one-dimensional periodic lattice that results in a rich phase diagram with the existence of a nonequilibrium tricritical point. Second, we consider a system of identical Kuramoto oscillators on a two-dimensional periodic square lattice. Relaxation dynamics in such a system is studied when the oscillators interact only with their nearest neighbors. For a random initial condition, the system ends up either in a completely phase-synchronized state or a phase-locked state characterized by the presence of topological defects in the phase-field of the oscillators. Finally, we investigate the effect of stochasticity on emergent behavior in such a system. In the presence of dichotomous noise as a stochastic force, the dynamics exhibits for any non-zero finite noise correlation time a nonequilibrium Berezinskii–Kosterlitz–Thouless (BKT)-like transition as one tunes the noise strength, similar to the equilibrium one in the presence of Gaussian white noise.