  In the present paper, we give a condensed review, for the nonspecialist
reader, of a new modelling framework for spatio-temporal processes, based on
L\'{e}vy theory. We show the potential of the approach in stochastic geometry
and spatial statistics by studying L\'{e}vy-based growth modelling of planar
objects. The growth models considered are spatio-temporal stochastic processes
on the circle. As a by product, flexible new models for space--time covariance
functions on the circle are provided. An application of the L\'{e}vy-based
growth models to tumour growth is discussed.
