Exploring gluon saturation, as a consequence of nonlinear gluon interaction at small value of Bjorken x in hadron and nucleus, is a challenging subject for QCD study. Although capturing a clear signature of the onset of the gluon saturation in hadrons and nuclei should be anticipated in a future EIC, current high energy proton-nucleus (pA) collisions also provide unique observables to probe into the underlying gluon saturation dynamics in target nuclei. Especially, inclusive hadron and heavy quarkonium productions at forward rapidity in pA collisions have been considered to be ideal probes for investigating the gluon saturation since hard scatterings involve small-x gluons. In this talk, I will briefly show an up-to-date review of inclusive hadron and heavy quarkonium productions in small-x formalism. I will also illustrate the following topical issues: (i) Comparison of first full NLO calculations for inclusive hadron production in pA collisions with data. (ii) Implementation of Sudakov factor (CSS evolution effect) for heavy quarkonium production in small-x formalism.