A comprehensive study of the symmetric Lévy stable probability density function is presented. This is performed for orders both less than 2, and greater than 2. The latter class of functions are traditionally neglected because of a failure to satisfy non-negativity. The complete asymptotic expansions of the symmetric Lévy stable densities of order greater than 2 are constructed, and shown to exhibit intricate series of transcendentally small terms-asymptotics beyond all orders. It is demonstrated that the symmetric Lévy stable densities of any arbitrary rational order can be written in terms of generalized hypergeometric functions, and a number of new special cases are given representations in terms of special functions. A link is shown between the symmetric Lévy stable density of order 4, and Pearcey's integral, which is used widely in problems of optical diffraction and wave propagation. This suggests the existence of applications for the symmetric Lévy stable densities of order greater than 2, despite their failure to define a probability density function.