In this video from 1981, David D. Friedman lists some unresolved problems with libertarianism and also tells some funny stories about Ludwig von Mises, Murray Rothbard, and Ayn Rand. In general, he thinks that libertarians are too confident in their ability to answer all real world problems given the current state of libertarian ethical theory. For anyone interested in the fundamentals of libertarianism, this video will be a lot of fun.
TL;DW: First, he is concerned that there is no pre-defined rule for quantifying the kind and quantity of punishment and restitution that is appropriate in response to crimes. Second, he wonders how accidental aggression should be handled compared to intentional aggression. Third, he wonders whether the conclusion that it is okay to shoot a human shield gives an ethical justification for any kind of aggression in order to protect oneself, including taxes and the draft. Fourth, he brings up the idea of absolute property and the trespassing photon problem. Fifth, he wonders how much risk to other people is acceptable when taking actions like flying a plane. Sixth, he laments that there is no solution to the question of justifying land ownership after so much theft of land has occurred. Seventh, he says that there is no clear business model for protecting people against nuclear attacks, and insinuates that the same issue applies to some other public goods.
Bonus problem: if the only choice is between surrender to the soviets and taxing your neighbors, what can a libertarian do?