  We present a step by step introduction to the notion of time-delay in
classical and quantum mechanics, with the aim of clarifying its foundation at a
conceptual level. In doing so, we motivate the introduction of the concepts of
"fuzzy" and "free-flight" sojourn times, that we use to provide the most
general possible definition for the quantum time-delay, valid for simple and
multichannel scattering systems, with or without conditions on the observation
of the scattering particle, and for incoming wave packets whose energy can be
smeared out or sharply peaked (fixed energy). We conclude our conceptual
analysis by presenting what we think is the right interpretation of the
concepts of sojourn and delay times in quantum mechanics, explaining why, in
ultimate analysis, they shouldn't be called "times".
