The Open Radio Access Network (RAN) is a networking paradigm that builds on top of cloud-based, multi-vendor, open and intelligent architectures to shape the next generation of cellular networks for 5G and beyond. While this new paradigm comes with many advantages in terms of observatibility and reconfigurability of the network, it inevitably expands the threat surface of cellular systems and can potentially expose its components to several cyber attacks, thus making securing O-RAN networks a necessity. In this paper, we explore the security aspects of O-RAN systems by focusing on the specifications and architectures proposed by the O-RAN Alliance. We address the problem of securing O-RAN systems with a holistic perspective, including considerations on the open interfaces used to interconnect the different O-RAN components, on the overall platform, and on the intelligence used to monitor and control the network. For each focus area we identify threats, discuss relevant solutions to address these issues, and demonstrate experimentally how such solutions can effectively defend O-RAN systems against selected cyber attacks. This article is the first work in approaching the security aspect of O-RAN holistically and with experimental evidence obtained on a state-of-the-art programmable O-RAN platform, thus providing unique guideline for researchers in the field.