  We derive the non-perturbative worldsheet S matrix for fundamental
excitations of Type IIB superstring theory on AdS_3 x S^3 x T^4 with
Ramond-Ramond flux. To this end, we study the off-shell symmetry algebra of the
theory and its representations. We use these to determine the S matrix up to
scalar factors and we derive the crossing equations that these scalar factors
satisfy. Our treatment automatically includes fundamental massless excitations,
removing a long-standing obstacle in using integrability to study the
AdS_3/CFT_2 correspondence. The present paper contains a detailed derivation of
results first announced in arXiv:1403.4543.
