Conjugations, (antilinear isometric involutions), in space $L^2$ of the unit circle commuting with multiplication by $z$ or intertwining multiplications by $z$ and $\bar z$ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces. We characterize all conjugations between two model spaces. Next, $L^2$ space on the real line will be considered with respect to conjugations. All conjugations invariant for multiplication operators or translation will be characterized.