We study continuous maps of metric spaces for which two nearby orbits are asymptotic (termed asymptotically expansive for short). We also analyze the bi-asymptotically expansive homeomorphisms namely asymptotically expansive homeomorphisms with asymptotically expansive inverse. Indeed, we obtain necessary and sufficient conditions for asymptotic expansivity, characterize the asymptotically expansive homeomorphisms of the circle, prove a spectral decomposition theorem and estimate the entropy through the growth rate of the periodic orbits.