We discuss the geometric properties of the quasihyperbolic metric.The quasihyperbolic metric is natural generalization of the hyperbolic metric originating from the work of F.W.Gehring and his students on quasiconformal mappings in 1970s.Since its introduction,this metric has found numerous applications in geometric function theory and related topics such as the potential theory.Despite these advances,many fundamental properties of the quasihyperbolic metric remain open questions.In this presentation,we investigate balls and geodesics in a domains of a eucludean and Banach spaces with quasihyperbolic metric.In particular,we consider(non-)existence of the geodesics,and smoothness of the geodesics and quasihyperbolic balls.This presentation is based on(ongoing)joint research with Riku Klén(University of Turku)and Jarno Talponen(University of Eastern Finland).