  We derive the necessary and sufficient conditions for a 2d QCD theory of
massless gluons and left and right chiral quarks in arbitrary representations
of a gauge group G to develop a mass gap. These results are obtained from
spectral properties of the lightcone and temporal QCD Hamiltonians. The
conditions can be explicitly solved, and we provide the complete list of all 2d
QCD theories that have a quantum mechanical gap in the spectrum, while any
other theory not in the list is gapless. The list of gapped theories includes
QCD models with quarks in vector-like as well as chiral representations. The
gapped theories consist of several infinite families of classical gauge groups
with quarks in rank 1 and 2 representations, plus a finite number of isolated
cases. We also put forward and analyze the effective infrared description of
QCD - TQFTs for gapped theories and CFTs for gapless theories - and exhibit
several interesting features in the infrared, such as the existence of
non-trivial global 't Hooft anomalies and emergent supersymmetry. We identify
2d QCD theories that flow in the infrared to celebrated CFTs such as minimal
models, bosonic and supersymmetric, and Wess-Zumino-Witten and Kazama-Suzuki
models.
