TITLE: Are there standard examples of stable theories that are undecidable?
QUESTION [9 upvotes]: What is known about decidability of various first order theories studied in stable model theory, geometric model theory, o-minimality?  For example, is there a natural example of an undecidable first order theory with nice stability properties? 

Many standard examples, such as ACF and hyperbolic groups, seem to decidable.
What about differentially closed fields ? Zariski geometries? What about theories obtained 
via Hrushovski fusion? 
I should add that there artificially constructed examples of non-decidable stable theories: say add infinitely many algebraic types coding an undecidable set.

REPLY [12 votes]: Let X be your favorite undecidable set of primes.  In a language with one unary function symbol, say f^p(x) = x iff p is in X  and also that there is a unique cycle of length p. Add that f is a bijection.  Now the prime model contains cycles of exactly the lengths in X and the theory is undecidable. But it is categorical in all uncountable power and therefore omega stable.