Exponential periods form a class of complex numbers containing the special values of the gamma and the Bessel functions, the Euler constant and other interesting numbers which are not expected to be periods in the usual sense. However, they appear as coefficients of the comparison isomorphism between two cohomology theories associated to varieties with a regular function: the de Rham cohomology of a connection with irregular singularities and the so-called “rapid decay” cohomology. I will explain how this point of view allows one to construct a Tannakian category of exponential motives and to produce Galois groups which conjecturally govern all algebraic relations among these numbers. The focus will be on examples and open questions rather than on the more abstracts aspects of the theory. This is a joint work with Peter Jossen.