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We prove in one go that each of the 4 families of Burniat surfaces with K^2 =6,5,4, is a connected component of the moduli space of surfaces of generaltype.  We prove also the rationality of each component. In the nodal case (one ofthe two families for K^2_S = 4) a very surprising and new phenomenon occurs.  Both the moduli space for the minimal models and the Gieseker moduli spacefor canonical models are everywhere non reduced.  But the nilpotence order is higher for the first.

Burniat surfaces II: secondary Burniat surfaces form three connected components of the moduli space