Today is the era of the Internet of Things (IoT). The recent advances in hardware and information technology have accelerated the deployment of billions of interconnected, smart and adaptive devices in critical infrastructures like health, transportation, environmental control, and home automation. Transferring data over a network without requiring any kind of human-to-computer or human-to-human interaction, brings reliability and convenience to consumers, but also opens a new world of opportunity for intruders, and introduces a whole set of unique and complicated questions to the field of Digital Forensics. Although IoT data could be a rich source of evidence, forensics professionals cope with diverse problems, starting from the huge variety of IoT devices and non-standard formats, to the multitenant cloud infrastructure and the resulting multi-jurisdictional litigations. A further challenge is the end-to-end encryption which represents a trade-off between users' right to privacy and the success of the forensics investigation. Due to its volatile nature, digital evidence has to be acquired and analyzed using validated tools and techniques that ensure the maintenance of the Chain of Custody. Therefore, the purpose of this paper is to identify and discuss the main issues involved in the complex process of IoT-based investigations, particularly all legal, privacy and cloud security challenges. Furthermore, this work provides an overview of the past and current theoretical models in the digital forensics science. Special attention is paid to frameworks that aim to extract data in a privacy-preserving manner or secure the evidence integrity using decentralized blockchain-based solutions. In addition, the present paper addresses the ongoing Forensics-as-a-Service (FaaS) paradigm, as well as some promising cross-cutting data reduction and forensics intelligence techniques. Finally, several other research trends and open issues are presented, with emphasis on the need for proactive Forensics Readiness strategies and generally agreed-upon standards.