It is natural and efficient to use human natural language (NL) directly to instruct robot task executions without prior user knowledge of instruction patterns. Currently, NL-instructed robot execution (NLexe) is employed in various robotic scenarios, including manufacturing, daily assistance, and health caregiving. It is imperative to summarize the current NLexe systems and discuss future development trends to provide valuable insights for upcoming NLexe research. This review categorizes NLexe systems into four types based on the robot’s cognition level during task execution: NL-based execution control systems, NL-based execution training systems, NL-based interactive execution systems, and NL-based social execution systems. For each type of NLexe system, typical application scenarios with advantages, disadvantages, and open problems are introduced. Then, typical implementation methods and future research trends of NLexe systems are discussed to guide the future NLexe research.