  Macdonald processes are measures on sequences of integer partitions built
using the Cauchy summation identity for Macdonald symmetric functions. These
measures are a useful tool to uncover the integrability of many probabilistic
systems, including the Kardar-Parisi-Zhang (KPZ) equation and a number of other
models in its universality class. In this paper we develop the structural
theory behind half-space variants of these models and the corresponding
half-space Macdonald processes. These processes are built using a Littlewood
summation identity instead of the Cauchy identity, and their analysis is
considerably harder than their full-space counterparts.
  We compute moments and Laplace transforms of observables for general
half-space Macdonald measures. Introducing new dynamics preserving this class
of measures, we relate them to various stochastic processes, in particular the
log-gamma polymer in a half-quadrant (they are also related to the stochastic
six-vertex model in a half-quadrant and the half-space ASEP). For the polymer
model, we provide explicit integral formulas for the Laplace transform of the
partition function. Non-rigorous saddle point asymptotics yield convergence of
the directed polymer free energy to either the Tracy-Widom GOE, GSE or the
Gaussian distribution depending on the average size of weights on the boundary.
