The present chapter is devoted to a rigorous treatment of the ’sums over paths’ proposed by Feynman as an alternative calculus for quantum mechanics. We begin with a formulation of the problem and a brief survey of some approaches to it. In Section 2 we study a theory in which the path integrals under consideration are defined by a Parseval-type equality, and refer to the functions on an abstract Hilbert space of paths. The latter is specified in Section 3: we discuss here various ways in which the path spaces referring to a quantum-mechanical system may be characterized. The most popular, at least among physicists, are the methods which follow Feynman’s original suggestion and define the path integral through the approximations based on piecewise linear paths. Section 4 is devoted to a detailed discussion of these methods. In particular, they allow us to extend the definition given in Section 2. Another ‘sequential’ definition of the Feynman path integral based on the product formulae is given in Section 5. The final section of this chapter contains more information about the other F-integral theories and their mutual relations.