  We show that every free amalgamation class of finite structures with
relations and (symmetric) partial functions is a Ramsey class when enriched by
a free linear ordering of vertices. This is a common strengthening of the
Ne\v{s}et\v{r}il-R\"odl Theorem and the second and third authors' Ramsey
theorem for finite models (that is, structures with both relations and
functions). We also find subclasses with the ordering property. For languages
with relational symbols and unary functions we also show the extension property
for partial automorphisms (EPPA) of free amalgamation classes. These general
results solve several conjectures and provide an easy Ramseyness test for many
classes of structures.
