Proton-nucleus (p+A) collisions play an important role as a control system for interpreting hot nuclear matter effects in nucleus-nucleus (A+A) collisions. There is a large amount of data from both RHIC and the LHC that indicate that collective effects are also present in small systems. Understanding the origin of these effects is still incomplete, since a variety of models with very different underlying physics have been shown to describe p+A data. We present a comprehensive set of measurements of soft-physics observables and comparisons with theoretical models. These measurements include multi-particle correlations as a function of event multiplicity, v(2)(p(T)) and v(3)(p(T)) for inclusive charged particles at mid-rapidity, v(2)(p(T)) for hadrons at forward/backward rapidities, and the centrality and pseudorapidity dependence of inclusive v(2). The implications for the origin of collectivity in p+Au collisions at RHIC is discussed.