The Internet of Things paradigm envisions the pervasive interconnection and cooperation of smart things over the current and future Internet infrastructure. The Internet of Things is, thus, the evolution of the Internet to cover the real-world, enabling many new services that will improve people's everyday lives, spawn new businesses and make buildings, cities and transport smarter. Smart things allow indeed for ubiquitous data collection or tracking, but these useful features are also examples of privacy threats that are already now limiting the success of the Internet of Things vision when not implemented correctly. These threats involve new challenges such as the pervasive privacy-aware management of personal data or methods to control or avoid ubiquitous tracking and profiling. This paper analyzes the privacy issues in the Internet of Things in detail. To this end, we first discuss the evolving features and trends in the Internet of Things with the goal of scrutinizing their privacy implications. Second, we classify and examine privacy threats in this new setting, pointing out the challenges that need to be overcome to ensure that the Internet of Things becomes a reality.