  In this survey article we give basic introduction to the theory of quantum
families of maps. We begin with a general look at non-commutative (or
"quantum") topology. Then we formulate all our results in this language.
Existence of quantum families possessing special properties is discussed and we
show that these quantum spaces are canonically endowed with quantum semigroup
structures. Classical analogy is emphasized at each step and many examples are
described.
