Abstract In this note we give short review of known results and announce new results (see below Theorem 8 and Theorem 6 and its generalizations). In the first part of this review paper , we focus on ultrahyperbolic metric and Ahlfors lemmas and to the estimate opposite to Ahlfors-Schwarz lemma proved by the author(Theorem 5-6). The second part is devoted to Ahlfors-Schwarz lemma for harmonic-quasiregular maps and some results obtained in [AMM].