  Moir\'e patterns of transition metal dichalcogenide (TMD) heterobilayers have
proven to be an ideal platform to host unusual correlated electronic phases,
emerging magnetism, and correlated exciton physics. While the existence of
novel moir\'e excitonic states is established through optical measurements, the
microscopic nature of these states is still poorly understood, often relying on
empirically fit models. Here, combining large-scale first-principles GW-BSE
calculations and micro-reflection spectroscopy, we identify the nature of the
exciton resonances in WSe$_2$/WS$_2$ moir\'e superlattices, discovering a
surprisingly rich set of moir\'e excitons that cannot be even qualitatively
captured by prevailing continuum models. Our calculations reveal moir\'e
excitons with distinct characters, including modulated Wannier excitons and
previously unindentified intralayer charge-transfer excitons. Signatures of
these distinct excitonic characters are confirmed experimentally via the unique
carrier-density and magnetic-field dependences of different moir\'e exciton
resonances. Our study highlights the highly non-trivial exciton states that can
emerge in TMD moir\'e superlattices, and suggests novel ways of tuning
many-body physics in moir\'e systems by engineering excited-states with
specific spatial characters.
