  We study possible geometries and R-symmetry breaking patterns that lead to
globally BPS surface operators in the six dimensional $\mathcal{N}=(2,0)$
theory. We find four main classes of solutions in different subspaces of
$\mathbb{R}^6$ and a multitude of subclasses and specific examples. We prove
that these constructions lead to supersymmety preserving observables and count
the number of preserved supercharges. We discuss the underlying geometry,
calculate their anomalies and present analogous structures for the holographic
dual M2-branes in $AdS_7\times S^4$. We also comment on the dimensional
reduction of these observables to line and surface operators in 4d and 5d
theories. This rich spectrum of operators are proposed as the simplest and most
natural observables of this mysterious theory.
