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We prove the Ramsey property for classes of ordered structures with closuresand given local properties. This generalises earlier results: theNe\v{s}et\v{r}il-R\"odl Theorem, the Ramsey property of partial orders andmetric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We usethis framework to solve several open problems and give new examples of Ramseyclasses. Among others, we find Ramsey lifts of convexly ordered $S$-metricspaces and prove the Ramsey theorem for finite models (i.e. structures withboth functions and relations) thus providing the ultimate generalisation of thestructural Ramsey theorem. Both of these results are natural, and easy tostate, yet their proofs involve most of the theory developed here.  We also characterise Ramsey lifts of classes of structures defined byfinitely many forbidden homomorphisms and extend this to special cases ofclasses with closures. This has numerous applications. For example, we findRamsey lifts of many Cherlin-Shelah-Shi classes.

All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)