The biomedical applications of fractal concepts have led to a wealth of new insights in biology and physiology, including a new formulation of the concept of health. Herein we review how over the past decade the complexity inherent in physiological structures and processes has been described by random fractals. In particular we argue that the scaling observed in these biological data sets is a consequence of the Lévy statistics of the underlying processes. This review interleaves arguments from renormalization group theory, fractal scaling and Lévy stable distributions in various physiological contexts including the mammalian lung, the beating of the heart, DNA sequencing, the dendritic branching of neurons and blood vessels, the dynamics of proteins, ion channel gating and radioactive clearance curves from the body in order to reveal an underlying unity to physiological processes. Wiebel promotes the use of fractal geometry as a design principle for living organisms, herein we suggest that Lévy stable statistics in being more inclusive may be an even more useful design principle.