  In this review-type paper written at the occasion of the Oberwolfach workshop
{\em One-sided vs. Two-sided stochastic processes} (february 22-29, 2020), we
discuss and compare Markov properties and generalisations thereof in more
directions, as well as weaker forms of conditional dependence, again either in
one or more directions. In particular, we discuss in both contexts various
extensions of Markov chains and Markov fields and their properties, such as
$g$-measures, Variable Length Markov Chains, Variable Neighbourhood Markov
Fields, Variable Neighbourhood (Parsimonious) Random Fields, and Generalized
Gibbs Measures.
