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Abstract: Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness -- that all "in" and "out" states lie in the standard (zero memory) Fock space -- and therefore leads to infrared (IR) divergences in the standard $S$-matrix amplitudes. In this paper we define an infrared finite scattering theory which assumes only (1) the existence of in/out algebras and (2) that Heisenberg evolution is an automorphism of these algebras. The resulting "superscattering" map $\$$ allows for transitions between different in/out memory states and agrees with the standard $S$-matrix when it is defined. We construct $\$$-amplitudes by defining (3) a "generalized asymptotic completeness" which accommodates states with memory in the space of asymptotic states and (4) a complete basis of improper states which generalize the usual $n$-particle momentum basis to account for states with memory. Using only general properties of $\$$, we prove an analog of the Weinberg soft theorems in quantum gravity and QED which imply that all $\$$-amplitudes are well-defined in the infrared. We comment on how one must generalize this framework to consider $\$$-amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories).