Matérn Gaussian fields (MGFs) have been popular modeling choices in many aspects of Bayesian methodologies. In this talk we will discuss a generalization of MGFs to manifolds and graphs. In the first part, we formalize the definition of MGFs on manifolds and introduce a sparse approximation based on graph discretization together with a convergence analysis, complementing the finite element approximations of MGFs on Euclidean domains. Numerical examples will demonstrate their wide applicability. In the second part, we introduce a framework for choosing the number of discretization nodes when approximating MGFs and demonstrate the ideas through the finite element approaches. We propose a conceptually simple selection rule and show that in many practical scenarios a low-rank finite element approximation is sufficient to further reduce computational cost without hindering the estimation accuracy.