  We present a hypothesis for the universal properties of operators evolving
under Hamiltonian dynamics in many-body systems. The hypothesis states that
successive Lanczos coefficients in the continued fraction expansion of the
Green's functions grow linearly with rate $\alpha$ in generic systems, with an
extra logarithmic correction in 1d. The rate $\alpha$ --- an experimental
observable --- governs the exponential growth of operator complexity in a sense
we make precise. This exponential growth even prevails beyond semiclassical or
large-$N$ limits. Moreover, $\alpha$ upper bounds a large class of operator
complexity measures, including the out-of-time-order correlator. As a result,
we obtain a sharp bound on Lyapunov exponents $\lambda_L \leq 2 \alpha$, which
complements and improves the known universal low-temperature bound $\lambda_L
\leq 2 \pi T$. We illustrate our results in paradigmatic examples such as
non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models.
Finally we use the hypothesis in conjunction with the recursion method to
develop a technique for computing diffusion constants.
