We show with computer simulations that a rich variety of static and dynamical colloidal phases can be realized for colloids interacting with two-dimensional periodic substrates. For the static case a new type of colloidal state that we term colloidal molecular crystals occurs when there is an integer number of colloids per substrate minima. Here there is a novel orientational ordering in addition to the positional ordering of the colloids. The colloidal molecular crystals exhibit a multi-step melting in which the orientational ordering is lost first, followed by the positional ordering. This multi-step melting phenomenon agrees well with recent experiments. Additionally we show that at fillings where the number of colloids is an incommensurate fraction of the number of substrate minima, as a function of temperature there is a transition to a state in which local incommensurations become thermally activated. With an applied drive we find that a remarkable number of distinct dynamical phases can be realized, including ordered and disordered flows. We also illustrate flow phases in which the colloidal motion locks to a symmetry direction of the underlying lattice.
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