  Isolated quantum systems with quenched randomness exhibit many-body
localization (MBL), wherein they do not reach local thermal equilibrium even
when highly excited above their ground states. It is widely believed that
individual eigenstates capture this breakdown of thermalization at finite size.
We show that this belief is false in general and that a MBL system can exhibit
the eigenstate properties of a thermalizing system. We propose that localized
approximately conserved operators (l$^*$-bits) underlie localization in such
systems. In dimensions $d>1$, we further argue that the existing MBL
phenomenology is unstable to boundary effects and gives way to l$^*$-bits.
Physical consequences of l$^*$-bits include the possibility of an eigenstate
phase transition within the MBL phase unrelated to the dynamical transition in
$d=1$ and thermal eigenstates at all parameters in $d>1$. Near-term experiments
in ultra-cold atomic systems and numerics can probe the dynamics generated by
boundary layers and emergence of l$^*$-bits.
