  Quantum groups have been studied within several areas of mathematics and
mathematical physics. This has led to different approaches, each of them with
their own techniques and conventions. Starting with Hopf algebras, where there
is a general consensus, moving in the direction of topological quantum groups,
where there is no such consensus, it is easy to get lost. Not only many
difficulties have to be overcome, but also several choices must be made. The
way this is done is often confusing. Some choices even turn out to be rather
annoying. As an introductory lecture at the conference on Topological quantum
groups and Hopf algebras in 2016, we have explained these choices, difficulties
and annoyances, encountered on the road from Hopf algebras to topological
quantum groups. In these notes, we discuss more aspects of the development of
locally compact quantum groups. We not only explain some of these difficulties
in greater detail, but we also give background information about the different
steps, combined with some historical comments. We start with finite quantum
groups and continue with discrete quantum groups, compact quantum groups and
algebraic quantum groups. Multiplicative unitaries are an important side track
before we finally arrive at locally compact quantum groups. Along the way, we
also formulate some interesting remaining problems in the theory.
