  We introduce a randomized Hall-Littlewood RSK algorithm and study its
combinatorial and probabilistic properties. On the probabilistic side, a new
model --- the Hall-Littlewood RSK field --- is introduced. Its various
degenerations contain known objects (the stochastic six vertex model, the
asymmetric simple exclusion process) as well as a variety of new ones. We
provide formulas for a rich class of observables of these models, extending
existing results about Macdonald processes. On the combinatorial side, we
establish analogs of properties of the classical RSK algorithm: invertibility,
symmetry, and a "bijectivization" of the skew-Cauchy identity.
