We present an extensive study of symmetric Boolean functions, especially of their cryptographic properties. Our main result establishes the link between the periodicity of the simplified value vector of a symmetric Boolean function and its degree. Besides the reduction of the amount of memory required for representing a symmetric function, this property has some consequences from a cryptographic point of view. For instance, it leads to a new general bound on the order of resiliency of symmetric functions, which improves Siegenthaler's bound. The propagation characteristics of these functions are also addressed and the algebraic normal forms of all their derivatives are given. We finally detail the characteristics of the symmetric functions of degree at most 7, for any number of variables. Most notably, we determine all balanced symmetric functions of degree less than or equal to 7.