Continuum robots (CRs), owing to their compact structure, inherent compliance, and flexible deformation, have been widely applied in various fields. By coordinating multiple CRs to form collaborative continuum robots (CCRs), task adaptability, workspace, flexibility, load capacity, and operational stability can be further improved, thus offering significant advantages. In recent years, interest in this emerging field has grown steadily within the continuum-robotics community, accompanied by a consistent rise in related publications. By presenting a comprehensive overview of recent progress from different system-architecture levels, this survey provides a clear framework for research on CCRs. First, CCRs are classified into the three collaboration modes of separated collaboration, assistance collaboration, and parallel collaboration, with definitions provided. Next, advances in structural design, modeling, motion planning, and control for each mode are systematically summarized. Finally, current challenges and future opportunities for CCRs are discussed.