The aim of this note is to introduce for point processes in $\R^d$ the notions general position and reinforced general position, and to characterize these processes. As a consequence we show that Poisson processes $P_\rho$ with an infinite intensity measures $\rho$ are in general position iff $\rho$ is diffuse in the sense that any affine subspace of dimension $d - 1$ is a $\rho$-nullset. Moreover, $P_{\rho}$ is in reinforced general position iff in addition any $(d - 1)$-sphere is a $\rho$-nullset.