abstract: Membrane-bound cellular organelles perform many essential functions, among which the sorting and biochemical maturation of cellular components. Organelles along the secretory and endocytic path- ways are strongly out-of-equilibrium structures, which display large stochastic uctuations of com- position and shape resulting from inter-organelle exchange and enzymatic reactions. Understanding how the dierent molecular mechanisms controlling these processes are orchestrated to yield robust fluxes of matter and to direct particular components to particular locations within the cell is an out- standing problem of great interest for cell biologist, but also for physicists. In this talk, I will discuss a conceptual model of organelle biogenesis and maintenance that include vesicular exchange (bud- ding, transport, and fusion) and biochemical maturation, i.e. the change of identity of an organelle over time (early to late endosomes, cis to trans Golgi cisternae). I will show how the non-equilibrium steady-state of an organelle or a network of organelles may be varied in a controlled manner by modifying a limited number of coarse-grained parameters (essentially, the budding, fusion and maturation rates). In the context of the Golgi apparatus, our model yields important ndings relating the structure and dynamics of the Golgi, and the way cargo is transported across this organelle. in particular, we have found that the two archetypical model of Golgi dynamics: vesicular transport and cisternal maturation, can be seen as extreme cases of a much richer dynamics, which can be tuned by varying the rates of vesicular budding and fusion, and biochemical maturation.