End-user privacy in mobile telephony systems is nowadays of great interest because of the envisaged hyper-connectivity and the potential of the unprecedented services (virtual reality, machine-type communication, vehicle-to-everything, IoT, etc.) being offered by the new 5G system. This paper reviews the state of subscription privacy in 5G systems. As the work on 5G Release 15 -- the first full set of 5G standards -- has recently been completed, this seems to be an appropriate occasion for such a review. The scope of the privacy study undertaken is limited to the wireless part of the 5G system which occurs between the service provider's base station and the subscriber's mobile phone. Although 5G offers better privacy guarantees than its predecessors, this work highlights that there still remain significant issues which need rectifying. We undertook an endeavor to (i) compile the privacy vulnerabilities that already existed in the previous mobile telephony generations. Thereafter, (ii) the privacy improvements offered by the recently finalized 5G standard were aggregated. Consequently, (iii) we were able to highlight privacy issues from previous generations that remain unresolved in 5G Release 15. For completeness, (iv) we also explore new privacy attacks which surfaced after the publication of the 5G standard. To address the identified privacy gaps, we also present future research directions in the form of proposed improvements.