The telescope conjecture has been formulated in stable homotopy theory
by Bousfield and Ravenel, and later by Neeman for compactly generated
triangulated categories. I will report on recent progress: The
conjecture has been established in joint work with Benson and Iyengar
for the stable category of representations of any finite group. Also,
the conjecture holds for the derived category of any hereditary algebra
by a joint result with Stovicek.