High entropy alloys are one of the most exciting developments conceived in the materials science field in the last years. These novel advanced engineering alloys exhibit a unique set of properties, which include, among others, good mechanical performance under severe conditions in a wide temperature range and high microstructural stability over long time periods. Owing to the remarkable properties of these alloys, they can become expedite solutions for multiple structural and functional applications. Nevertheless, like any other key engineering alloy, their capacity to be welded, and thus become a permanent feature of a component or structure, is a fundamental issue that needs to be addressed to further expand these alloys’ potential applications. In fact, welding of high entropy alloys has attracted some interest recently. Therefore, it is important to compile the available knowledge on the current state of the art on this topic in order to establish a starting point for the further development of these alloys. In this article, an effort is made to acquire a comprehensive knowledge on the overall progress on welding of different high entropy alloy systems through a systematic review of both fusion-based and solid-state welding techniques. From the current literature review, it can be perceived that welding of high entropy alloys is currently gaining more interest. Several high entropy alloy systems have already been successfully welded. However, most research works focus on the well-known CoCrFeMnNi. For this specific system, both fusion and solid-state welding have been used, with no significant degradation of the joints’ mechanical properties. Among the different welding techniques already employed, laser welding is predominant, potentially due to the small size of its heat source. Overall, welding of high entropy alloys is still in its infancy, though good perspectives are foreseen for the use of welded joints based on these materials in structural applications.