  A number of recent results in Euclidean Harmonic Analysis have exploited
several adjacent systems of dyadic cubes, instead of just one fixed system. In
this paper, we extend such constructions to general spaces of homogeneous type,
making these tools available for Analysis on metric spaces. The results include
a new (non-random) construction of boundedly many adjacent dyadic systems with
useful covering properties, and a streamlined version of the random
construction recently devised by H. Martikainen and the first author. We
illustrate the usefulness of these constructions with applications to weighted
inequalities and the BMO space; further applications will appear in forthcoming
work.
