Given an algebraic curve C defined over Q, an involution \iota on C and a quadratic extension Q(\sqrt{d})/Q, we can form a new curve C_d, the "quadratic twist" of C. Under certain hypotheses on C and \iota, we can show that a positive proportion of twists by quadratic fields of prime discriminant have points everywhere locally but not globally. The hypotheses apply in particular to twists of the modular curves X_0(N) -- for squarefree N > 163 -- by the Atkin-Lehner involution w_N, as well as to certain Shimura curves. These twisted modular curves are of particular interest as moduli spaces of quadratic Q-curves and hence as repositories of homomorphisms from Gal_Q to PGL_2(F_p). It is interesting to study local and global points on these curves in a more systematic way. When X_0(N) has genus zero or one we have essentially complete answers, thanks to Shih, Serre and myself, and these results have applications to the realization of PSL_2(F_p) as a Galois group over Q.