Limit ultrapowers and abstract logics

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L. For every countably generated [cl, wo]-compact logic L, our main applications are: (i) Elementary classes of L can be characterized in terms of -L only. (ii) If W and 93 are countable models of a countable superstable theory without the finite cover property, then 1 _L 9-. (iii) There exists the "largest" logic M such that complete extensions in the sense of M and L are the same; moreover M is still [wo, wo]-compact and satisfies an interpolation property stronger than unrelativized A-closure. (iv) If L = L..(Q.), then cf(w) > co and x" < w2 for all A < way. We also prove that no proper extension of L.,<, generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning L,,-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.