It is known that in the theory of quantum groupoids of separable type (developed by the author and Van Daele), there arise certain multiplicative partial isometries that behave in a similar manner as the multiplicative unitaries (in the sense of Baaj-Skandalis) associated with a quantum group. In this talk, I will give and explain some algebraic conditions that axiomatically determine a multiplicative partial isometry $W$. Then we will also consider the manageability condition for $W$. Starting from the multiplicativity and the manageability, we can construct most of the quantum groupoid structure, including a C*-algebra $A$ with a comultiplication $\Delta$, base C*-algebras $B_s$ and $B_t$, as well as the antipode map and its polar decomposition. Its dual quantum groupoid can be also constructed. We will then turn to exploring an ongoing project (with Woronowicz) concerning the partial isometries that are "adapted" to our given manageable multiplicative unitary $W$.