We present a numerical study of dark matter halo concentrations in $Λ$CDM and self-similar cosmologies. We show that the relation between concentration, $c$, and peak height, $ν$, exhibits the smallest deviations from universality if halo masses are defined with respect to the critical density of the universe. These deviations can be explained by the residual dependence of concentration on the local slope of the matter power spectrum, $n$, which affects both the normalization and shape of the $c$-$ν$ relation. In particular, there is no well-defined floor in the concentration values. Instead, the minimum concentration depends on redshift: at fixed $ν$, halos at higher $z$ experience steeper slopes $n$, and thus have lower minimum concentrations. We show that the concentrations in our simulations can be accurately described by a universal seven-parameter function of only $ν$ and $n$. This model matches our $Λ$CDM results to $\lesssim 5\%$ accuracy up to $z = 6$, and matches scale-free $Ω_{\rm m} = 1$ models to $\lesssim 15\%$. The model also reproduces the low concentration values of Earth--mass halos at $z ≈ 30$, and thus correctly extrapolates over $16$ orders of magnitude in halo mass. The predictions of our model differ significantly from all models previously proposed in the literature at high masses and redshifts. Our model is in excellent agreement with recent lensing measurements of cluster concentrations. ; Comment: Published version, minor changes. 17 pages, 14 figures. The corresponding public python code can be found at http://www.benediktdiemer.com/code