In recent decades, the urgency to protect and upgrade cultural heritage structures (CHS) has become of primary importance due to their unique value and potential areas of impact (economic, social, cultural, and environmental). Structural health monitoring (SHM) and the management of CHS are emerging as decisive safeguard measures aimed at assessing the actual state of the conservation and integrity of the structure. Moreover, the data collected from SHM are essential to plan cost-effective and sustainable maintenance solutions, in compliance with the basic preservation principles for historic buildings, such as minimum intervention. It is evident that, compared to new buildings, the application of SHM to CHS is even more challenging because of the uniqueness of each monitored structure and the need to respect its architectural and historical value. This paper aims to present a state-of-the-art evaluation of the current traditional and innovative SHM techniques adopted for CHS and to identify future research trends. First, a general introduction regarding the use of monitoring strategies and technologies for CHS is presented. Next, various traditional SHM techniques currently used in CHS are described. Then, attention is focused on the most recent technologies, such as fibre optic sensors and smart-sensing materials. Finally, an overview of innovative methods and tools for managing and analysing SHM data, including IoT-SHM systems and the integration of BIM in heritage structures, is provided.