In this talk, I will report on my recent (and on-going) joint work with Alfons Van Daele on developing the definition of a locally compact (C*-algebraic) quantum groupoid. At the purely algebraic level, this is closely related with the notion of ``weak multiplier Hopf algebras'' by Van Daele and Wang. Here, the comultiplication map cannot be non-degenerate, so a special idempotent element E plays an important role.

Many of the techniques from the locally compact quantum group theory carry over, with some adjustments. However, there are some different challenges, concerning the canonical idempotent E (which would be 1 in the quantum group case), working with the Haar weights, and the like. As time permits, I will give some explanations on these aspects, as well as some possible future applications. The work is still on-going, but the expected finished version should be closely related with the notion of ``measured quantum groupoids'' by Enock, Lesieur, and others, while technically somewhat different.