In this chapter, we explore fractal wandering through the lens of a particularly notable example—Weierstrass wandering. These walks elucidate the fundamental mechanism behind the emergence of algebraically diminishing long-range “tails” in both probability distributions and correlation functions. As demonstrated, this phenomenon is rooted in rare, extreme events generated, e.g., by the stochastically self-similar (hierarchical) structure of the trajectory of the process (i.e., the wandering of a test particle). Viewed broadly, this trajectory exhibits a stochastic fractal structure. The presence of rare, extreme events introduces a novel avenue of exploration in statistical physics and chaotic dynamics. This path diverges from traditional approaches and, despite notable progress, remains an area of active and intense research. In this chapter, we employ the Weierstrass spatial function. In Chap.  6 , we considered the Weierstrass time function. From this perspective, Chaps. 6 and 7 are complementary, covering the space-time of random events.