The consensus among cultural evolutionists seems to be that human cultural evolution is cumulative, which is commonly understood in the specific sense that cultural traits, especially technological traits, increase in complexity over generations. Here we argue that there is insufficient credible evidence in favor of or against this technological complexity thesis. For one thing, the few datasets that are available hardly constitute a representative sample. For another, they substantiate very specific, and usually different versions of the complexity thesis or, even (...) worse, do not point to complexity increases. We highlight the problems our findings raise for current work in cultural-evolutionary theory, and present various suggestions for future research. (shrink)