The worldline formalism provides an alternative to Feynman diagrams in the construction of amplitudes and effective actions that shares some of the superior properties of the organization of amplitudes in string theory. In particular, it allows one to write down integral representations combining the contributions of large classes of Feynman diagrams of different topologies. However, calculating these integrals analytically without splitting them into sectors corresponding to individual diagrams poses a formidable mathematical challenge. We summarize the history and state of the art of this problem, including some natural connections to the theory of Bernoulli numbers and polynomials and multiple zeta values.