  We consider Fra\"iss\'e structures whose objects have finite big Ramsey
degree and ask what consequences this has for the dynamics of the automorphism
group. Motivated by a theorem of D. Devlin about the partition properties of
the rationals, we define the notion of a big Ramsey structure, a single
structure which codes the big Ramsey degrees of a given Fra\"iss\'e structure.
This in turn leads to the definition of a completion flow; we show that if a
Fra\"iss\'e structure admits a big Ramsey structure, then the automorphism
group admits a unique universal completion flow. We also discuss the problem of
when big Ramsey structures exist and explore connections to the notion of
oscillation stability defined by Kechris, Pestov, and Todor\v{c}evi\'c
