In this paper, we introduce Lévy driven Cox point processes (LCPs) as Cox point processes with driving intensity function $\Lambda$ defined by a kernel smoothing of a Lévy basis (an independently scattered infinitely divisible random measure). We also consider log Lévy driven Cox point processes (LLCPs) with $\Lambda$ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes and log shot noise Cox processes. We study the theoretical properties of Lévy based Cox processes, including moment properties described by nth order product densities, mixing properties, specification of inhomogeneity and spatio-temporal extensions.