The talk is concerned with the multivariate stochastic collocation methods on unstructured grids. The motivation for such a study is the applications in parametric Uncertainty Quantification (UQ). We shall first give a general framework of stochastic collocation methods, which include approaches such as compressed sensing, least-squares, and interpolation. Particular attention will be then given to the least-squares approach, and we will review recent progresses. The talk is based on joint works with A. Narayan (Umass, USA), J. Jackman (Sandia, USA), Z. Xu (AMSS, CAS)