Let $mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{ eq}0$. A curve ${alpha}$ immersed in $mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${eta}$ and a one-to-one correspondence between ${alpha}$ and ${eta}$ 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 $mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${lambda}{ eq}0$ and ${mu}$ such that ${lambda}{kappa}+{mu}{ au}=1$, where ${kappa}$ and ${ au}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $mathbb{M}^3_q(c)$ are the only twisted curves in $mathbb{M}^3_q(c)$ 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.