The milling process of aerospace thin-walled parts requires extremely high geometric precision and surface quality, as these factors significantly influence aircraft performance and operational reliability. Milling chatter not only severely compromises machined surface integrity and accelerates tool wear, but also induces catastrophic production failures and significant economic losses. Over recent decades, the machining community has dedicated substantial efforts to investigating milling chatter mechanisms and developing corresponding control strategies. Remarkable progress has been made in terms of chatter stability prediction, online condition monitoring, and active/passive suppression techniques, with the ultimate objective of achieving chatter-free milling operations. However, compared with conventional milling processes, thin-walled part machining presents distinctive challenges due to their inherent characteristics such as low structural rigidity, poor machinability, and complex dynamics involved during milling operations (including time-varying behaviors, modal coupling, and position-dependent effects). These combined factors pose significant obstacles to effective chatter control. This paper consequently concentrates on recent advancements in milling chatter research for aerospace thin-walled parts: (i) Establishing dynamic models that accurately characterize actual milling processes by incorporating force-induced deformation and tool wear effects; integrating dynamic parameter updating techniques with probabilistic stability lobe diagram (SLD) solution approaches to provide risk-aware chatter prediction results. (ii) Leveraging multi-signal fusion and statistical analysis/artificial intelligence (AI) to realize real-time chatter condition monitoring; exploring effective measures to improve monitoring model generalization capabilities under limited sample sizes and variable operational conditions. (iii) Evaluating passive and active chatter suppression strategies systematically, combined with digital twin technology to enable seamless integration of chatter monitoring, suppression, and process optimization. (iv) Discussing milling chatter-induced part surface/sub-surface defects, with related indexes to quantify the effect of chatter marks on surface integrity. Through critical analysis of cutting-edge research and industrial applications, we further evaluate current research limitations and present promising future directions. These include innovations in chatter mechanism modeling, uncertainty quantification, physics-AI hybrid methodologies, edge-cloud-fog monitoring systems, novel materials development, metaverse-enabled human-computer interfaces, and collaborative control technologies of shape accuracy-surface integrity.