Abstract
Besides constituting an important and often crucial tool in studying semigroup varieties and pseudovarieties, relatively free and free profinite semigroups have proved to be of interest from many other natural standpoints. We present three groups of new problems concerning certain relatively free and free profinite semigroups and give background and motivation for them. The problems deal with the first-order theories of free profinite semigroups, the structure of free Burnside semigroups with commuting idempotents and of free Burnside inverse semigroups, and the representation of finitely generated free bands by order preserving transformations of a finite chain.
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