Numbers and functions in quantum field theory

We review recent results in the theory of numbers and single-valued functions on the complex plane which arise in quantum field theory. These results are the basis for a new approach to high-loop-order calculations. As concrete examples, we provide scheme-independent counterterms of primitive log-divergent graphs in ${\ensuremath{\phi}}^{4}$ theory up to eight loops and the renormalization functions $\ensuremath{\beta}$, $\ensuremath{\gamma}$, ${\ensuremath{\gamma}}_{m}$ of dimensionally regularized ${\ensuremath{\phi}}^{4}$ theory in the minimal subtraction scheme up to seven loops.