In this paper we consider functions analytic in a given finitely connected domain apart from a finite number of singularities of possibly logarithmic type. We prove some area theorems which generalize to these functions certain known results, in particular Goluzin's area theorem on functions $p$ -valent in a disc. We establish some integral criteria of when functions meromorphic in a given multiply-connected domain are univalent functions there and pairwise without common values.