In this paper, we report our recent advances on vertex centered finite volume element methods (FVEMs) for second order partial differential equations (PDEs). We begin with a brief review on linear and quadratic finite volume schemes. Then we present our recent advances on finite volume schemes of arbitrary order. For each scheme, we first explain its construction and then perform its error analysis under both H 1 and L 2 norms along with study of superconvergence properties.