In the theory of (C*- or vN-algebraic) locally compact quantum groups and quantum groupoids, the antipode (or coinverse) map is typically not part of the defining axioms. Rather, it is constructed from the existence of the left and right invariant weights. In this talk, we will discuss how the construction of the antipode map is carried out, and how it is defined in terms of its polar decomposition. We will mostly consider the quantum group case, but the quantum groupoid case will be also mentioned, by pointing out where the similarities and differences are.