Hurewicz's conjecture that Menger spaces are sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for other definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces are sigma-compact, but projections of co-analytic spaces need not be. The co-analytic case may be undecidable, but is not yet settled. We have partial results. Our methods are a mix of non-metric descriptive set theory and Arhangel'skii's work on generalized metric spaces.