  A long-standing theoretical prediction is that in clean, nodal unconventional
superconductors the magnetic penetration depth $\lambda$, at zero temperature,
varies linearly with magnetic field. This non-linear Meissner effect is an
equally important manifestation of the nodal state as the well studied
linear-in-$T$ dependence of $\lambda$, but has never been convincingly
experimentally observed. Here we present measurements of the nodal
superconductors CeCoIn$_5$ and LaFePO which clearly show this non-linear
Meissner effect. We further show how the effect of a small dc magnetic field on
$\lambda(T)$ can be used to distinguish gap nodes from non-nodal deep gap
minima. Our measurements of KFe$_2$As$_2$ suggest that this material has such a
non-nodal state.
