  Exceptional points (EPs), at which both eigenvalues and eigenvectors
coalesce, are ubiquitous and unique features of non-Hermitian systems.
Second-order EPs are by far the most studied due to their abundance, requiring
only the tuning of two real parameters, which is less than the three parameters
needed to generically find ordinary Hermitian eigenvalue degeneracies.
Higher-order EPs generically require more fine-tuning, and are thus assumed to
play a much less prominent role. Here, however, we illuminate how physically
relevant symmetries make higher-order EPs dramatically more abundant and
conceptually richer. More saliently, third-order EPs generically require only
two real tuning parameters in the presence of either a parity-time (PT)
symmetry or a generalized chiral symmetry. Remarkably, we find that these
different symmetries yield topologically distinct types of EPs. We illustrate
our findings in simple models, and show how third-order EPs with a generic
$\sim k^{1/3}$ dispersion are protected by PT symmetry, while third-order EPs
with a $\sim k^{1/2}$ dispersion are protected by the chiral symmetry emerging
in non-Hermitian Lieb lattice models. More generally, we identify stable, weak,
and fragile aspects of symmetry-protected higher-order EPs, and tease out their
concomitant phenomenology.
