The article discusses the role of non-Born-Oppenheimer couplings in dissociative recombination (DR) of triatomic and diatomic molecular ions. For highly symmetric molecular ions the couplings play an important role, which results in a high rate for the indirect DR process. The non-Born-Oppenheimer couplings are particularly strong in highly-symmetric ions owing to the degeneracy of electronic and vibrational states at the equilibrium configuration of the nuclei. This is a generic property of polyatomic molecular ions, frequently associated with the Jahn-Teller or Renner-Teller effect, although analogous situations should occur in other species having conical intersections. In this respect, DR in triatomic ions differs significantly from DR in diatomic systems, where indirect DR is usually slow, although one exception to this rule is reviewed here. We concentrate here on the dissociative recombination of two highly symmetric triatomic ions: H⁺₃ and HCO⁺. In both ions, non-adiabatic couplings between doubly-degenerate vibrational and doubly-degenerate electronic states play an important role and produce a fast indirect DR. We also compare DR in these triatomic ions with the exceptional diatomic ion LiH⁺, where non-adiabatic effects are strong due to l-mixing between p and d waves and this contributes to a high indirect DR rate. For all the systems considered, good agreement with experiments has been achieved, although some discrepancies remain to be resolved in the future.