Nowadays, robots in the modern world are playing an important and increasingly popular role. MRA (Mobile Robotic Assistant) is a type of mobile robot designed to support humans in many different fields, helping to improve efficiency and safety in daily activities, work, or medical treatment. The number of MRAs is increasing and diverse in function, in addition to the ability to collect and process data, MRAs also have the ability to physically interact with users. Therefore, security is one of the important issues to improve the safety and effective operation of MRA. In this paper, through a comprehensive literature review and detailed analysis of the prominent MRA security attacks in recent years (based on criteria such as: attack targets, technologies used, impact level, feasibility, and contribution to addressing overall MRA security issues), a systematic classification by MRA activity fields is conducted. Security attacks, threats, and vulnerabilities are examined from various perspectives, such as hardware attacks or network/system-level attacks, operating systems/application software. Additionally, corresponding security solutions are proposed, compared, and evaluated to enhance MRA security. The paper also addresses challenges and suggests open research directions for the future.