Modelling and Simulation as a Service (MSaaS) is a promising approach to deploy and execute Modelling and Simulation (M&S) applications quickly and on-demand. An appropriate software architecture is essential to deliver quality M&S applications following the MSaaS concept to a wide range of users. This study aims to characterize the state-of-the-art MSaaS architectures by conducting a systematic review of 31 papers published from 2010 to 2018. Our findings reveal that MSaaS applications are mainly designed using layered architecture style, followed by service-oriented architecture, component-based architecture, and pluggable component-based architecture. We also found that interoperability and deployability have the greatest importance in the architecture of MSaaS applications. In addition, our study indicates that the current MSaaS architectures do not meet the critical user requirements of modern M&S applications appropriately. Based on our results, we recommend that there is a need for more effort and research to (1) design the user interfaces that enable users to build and configure simulation models with minimum effort and limited domain knowledge, (2) provide mechanisms to improve the deployability of M&S applications, and (3) gain a deep insight into how M&S applications should be architected to respond to the emerging user requirements in the military domain.