Let

denote the 3-dimensional space form of index

, and constant curvature

. A curve

immersed in

is said to be a Bertrand curve if there exists another curve

and a one-to-one correspondence between

and

such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in

correspond with curves for which there exist two constants

and

such that

, where

and

stand for the curvature and torsion of the curve. As a consequence, non-null helices in

are the only twisted curves in

having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.