Advances in additive manufacturing (AM) processes have increased the number of relevant applications in various industries. To keep up with this development, the process stability of AM processes should be monitored, which is conducted through the assessment of the outputs or product characteristics. However, the use of univariate control charts to monitor an AM process might lead to misleading results, as most additively manufactured products have more than one correlated quality characteristic (QC). This paper proposes a framework for monitoring the multivariate quality characteristics of AM processes, and the proposed framework was applied to monitor a fused filament fabrication (FFF) process. In particular, specimens were designed and produced using the FFF process, and their QCs were identified. Then, critical quality characteristic data were collected using a precise measurement system. Furthermore, we propose a transformation algorithm to ensure the normality of the collected data. After examining the correlations between the investigated quality characteristics, a multivariate exponential weighted moving average (MEWMA) control chart was used to monitor the stability of the process. Furthermore, the MEWMA parameters were optimized using a novel heuristic technique. The results indicate that the majority of the collected data are not normally distributed. Consequently, the efficacy of the proposed transformation technique is demonstrated. In addition, our findings illustrate the correlations between the QCs. It is worth noting that the MEWMA optimization results confirm that the considered AM process (i.e., FFF) is relatively stable.