Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been worked on quite a bit since the time of writing. The list is very good as is, but there must also be a few good additions since 1999. Can someone point me to a more recent list of open problems in algebraic topology? My googling isn't turning up much else. Thank you!
I have made a note of some problems in the area of Nonabelian algebraic topology and homological algebra in 1990, and in Chapter 16 of the book in the same area and advertised here, with free pdf, there is a note of 32 problems and questions in this area which had occurred to me. These problems may well seem "narrow", and/or "out-of-line" of current trends, but I thought the latter big book ought not to be seen as any kind of final product, but hopefully as the beginning of a new story. Also it is interesting to record for any area of study what it might be hoped to do but so far does not.
Further comment: The origin of this work was methodological (does that count as a "problem"?). In the 1960s, writing the first edition of the book which is now "Topology and Groupoids" (2006), I convinced myself that all of $1$-dimensional homotopy theory was better expressed in terms of groupoids rather than groups. So the next question was: are groupoids useful, or not, or to what extent, in higher dimensional homotopy theory? Trying to find answers to this has been a lot of fun for all concerned, though the 1981 published work with Philip Higgins was described once to me by Michael Barratt as "Carved out of solid rock, and pursued in the teeth of opposition".
The website discussed in the comments went live a while back, and has a bunch of problems: