Additive manufacturing (AM), also known as 3D printing (3DP), refers to manufacturing technologies that build up the desired geometries by adding materials layer by layer. Common meltable and fusible materials such as polymers, metals, and ceramics could be used in 3DP processes. During decades of development, products made by 3DP can now achieve stringent industrial standards at comparable costs compared to those traditionally manufactured. Improving 3DP technologies is required to make them more competitive and acceptable than their counterparts. However, achieving this is challenging since the quality of printing products is still heavily dependent on many cost-driven factors. Inadequate quality, impaired functionality, and reduced service life are three main consequences of 3DP’s failures. To effectively detect and mitigate defects and failures of 3DP products, machine condition monitoring (MCM) technologies have been used to monitor 3D printing processes. With the help of those dedicated algorithms, it could also prevent failures from occurrence by alerting operators to take appropriate actions accordingly. This study systematically reviews the MCM technologies used in a typical 3DP process—the fused deposition modelling (FDM), identifying their advantages and disadvantages. The mentioned MCM technologies include but are not limited to traditional MCM (sensors only), aided with analytical and artificial intelligence (AI) tools. The MCM techniques focus on the defects of the 3DP process. The detection and identification of those defects are investigated. Furthermore, research trends on developing MCM technologies, including challenges and opportunities, are identified for improving the FDM process. This review highlights the developed methodologies of MCM that are applied to FDM processes to detect and identify abnormalities such as defects and failures. The evaluations of defects are elaborated to deepen the comprehension of the essence of the defects, including their cause, severity, and effect. A detailed deliberation about identifying the critical components for the successful application of 3DP MCM systems was done. Finally, this review indicates the technical barriers that need to be overcome to enhance the performance of monitoring, detection, and prediction by MCM and associated technologies.