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.
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