Autonomous UAV cinematography is an active research field with exciting potential for the media industry. It bears the promise of greatly facilitating UAV shooting for various applications, while significantly reducing the costs compared to manual shooting. However, the general problem has not been clearly defined and the challenges arising from current legislation and technology restrictions have not been fully charted. A complete overview of issues related to autonomous UAV cinematography is needed, pertaining to the current situation in the field, so as to guide immediate-future research. The purpose of this paper is to lay exactly this groundwork, with the expectation of providing a global perspective to multiple domain-specific research communities. The outlined issues are partitioned into challenges deriving from ethical/legal/safety considerations and from operational/production requirements. A brief survey of current technological solutions, including their limitations, is also provided for each issue.