We consider surfaces M immersed into Rnand we prove that the quantity ∫M|A|2(where A is the second fundamental form) controls in many ways the behaviour of conformal parametrizations of M. If M is complete, connected, noncompact and ∫M|A|2< ∞ we obtain a more or less complete picture of the behaviour of the immersions. In particular we prove that under these assumptions the immersions are proper. Moreover, if ∫M|A|2≤ 4π or if n = 3 and ∫M|A|2< 8π, then M is embedded. We also prove that conformal parametrizations of graphs of W2, 2functions on R2exist, are bilipschitz and the conformal metric is continuous. The paper was inspired by recent results of T.Toro.