In this thesis, we study some geometrical aspects of metric measure spaces (Rn, psi1/2 , mu)where mu is a locally finite regular Borel measure and a metric on psi1/2 which arises from a continuous negative definite function psi : Rn &rarr; R which satisfies psi(xi) &ge; 0 with psi(xi) = 0. This study is motivated by the investigation of a transition density estimate for pure jump processes on a general metric measure space. To gain a better insight into the behaviour of transition functions of symmetric Levy processes in this general setting, it seems desirable to understand geometrical properties of their underlying state spaces. More precisely, we show completeness of the metric spaces (Rn, psi1/2) and study under which circumstances open balls Bpsi(x,r), x &isin; Rn, r > 0, with respect to this metric are convex. Moreover, we focus on conditions of the metric measure spaces (Rn,psi1/2 ,mu) for the balls to satisfy the volume growth property [equation] for mu-almost all x &isin; Rn, 0 < r < R and a constant Cpsi(x,R)&ge;1. Finally, we show that the homogeneity property of a metric measure space can be applied to our case and provide some results associated with the construction of a Hajlasz-Sobolc space over (Rn,psi1/2, lambda(n)),where lambda(n) denotes the n-dirnensional Lebesgue measure.