This paper analytically derives the delay-capacity tradeoffs for Lévy mobility: Lévy walks and Lévy flights. Lévy mobility is a random walk with a power-law flight distribution. α is the power-law slope of the distribution and 0 <;; α ≤ 2. While in Lévy flight, each flight takes a constant flight time, in Lévy walk, it has a constant velocity which incurs strong spatio-temporal correlation as flight time depends on traveling distance. Lévy mobility is of special interest because it is known that Lévy mobility and human mobility share several common features including heavy-tail flight distributions. Humans highly influence the mobility of nodes (smartphones and cars) in real mobile networks as they carry or drive mobile nodes. Understanding the fundamental delay-capacity tradeoffs of Lévy mobility provides important insight into understanding the performance of real mobile networks. However, its power-law nature and strong spatio-temporal correlation make the scaling analysis non-trivial. This is in contrast to other random mobility models including Brownian motion, random waypoint and i.i.d. mobility which are amenable for a Markovian analysis. By exploiting the asymptotic characterization of the joint spatio-temporal probability density functions of Lévy models, the order of critical delay, the minimum delay required to achieve more throughput than Θ(1/√n) where n is the number of nodes in the network, is obtained. The results indicate that in Lévy walk, there is a phase transition that for 0 <; α <; 1, the critical delay is constantly Θ(n^1/2) and for 1 ≤ α ≤ 2, is Θ(n^α/2). In contrast, Lévy flight has critical delay Θ(n^α/2) for 0 <; α ≤ 2.