We theoretically study the few- and many-body dynamics of photons in chiral waveguides. In particular, we examine pulse propagation through an ensemble of N two-level systems chirally coupled to a waveguide. We show that the system supports correlated multiphoton bound states, which have a well-defined photon number n and propagate through the system with a group delay scaling as 1/n2. This has the interesting consequence that, during propagation, an incident coherent-state pulse breaks up into different bound-state components that can become spatially separated at the output in a sufficiently long system. For sufficiently many photons and sufficiently short systems, we show that linear combinations of n-body bound states recover the well-known phenomenon of mean-field solitons in self-induced transparency. Our work thus covers the entire spectrum from few-photon quantum propagation, to genuine quantum many-body (atom and photon) phenomena, and ultimately the quantum-to-classical transition. Finally, we demonstrate that the bound states can undergo elastic scattering with additional photons. Together, our results demonstrate that photon bound states are truly distinct physical objects emerging from the most elementary light-matter interaction between photons and two-level emitters. Our work opens the door to studying quantum many-body physics and soliton physics with photons in chiral waveguide QED. © 2020 authors.