  Magic-angle twisted bilayer graphene is the best studied physical platform
featuring moire potential induced narrow bands with non-trivial topology and
strong electronic correlations. Despite their significance, the Chern
insulating states observed at a finite magnetic field -- and extrapolating to a
band filling, $s$, at zero field -- remain poorly understood. Unraveling their
nature is among the most important open problems in the province of moir\'e
materials. Here we present the first comprehensive study of interacting
electrons in finite magnetic field while varying the electron density, twist
angle and heterostrain. Within a panoply of correlated Chern phases emerging at
a range of twist angles, we uncover a unified description for the ubiquitous
sequence of states with the Chern number $t$ for $(s,t)=\pm (0,4),
\pm(1,3),\pm(2,2)$ and $\pm(3,1)$. We also find correlated Chern insulators at
unconventional sequences with $s+t\neq \pm 4$, as well as with fractional $s$,
and elucidate their nature.
