These probabilistic objects are all defined in terms of families of probability kernels satisfying appropriate consistency relations. This common feature can be exploited to develop parallel theories in which concepts and techniques can be shared or imported. The talk will (i) present the main lines of this parallel treatment; (ii) discuss the resulting body of comparable properties (extremality and mixing, limit states, Dobrushin uniqueness criteria); (iii) review instances of interplay between the theories (phase transitions, uniqueness criteria), and (iv) present some remaining questions and open problems