Outphasing power amplifiers (OPAs) can be used in wireless information and power transfer systems due to their excellent conversion efficiency. With the rapid advancement of technology, there is a strong demand for OPAs to achieve wideband and multi-band operations to enhance frequency coverage. This paper provides a comprehensive review of the research progress on the development of broadband and multi-band board-level OPAs. Firstly, the paper introduces the fundamental components of an OPA. Then, the paper delves into a detailed analysis of the load modulation of OPAs. Subsequently, previous works on the circuit topology and design methods for broadband and multi-band OPAs are discussed and compared. The paper highlights various design strategies and their respective advantages and disadvantages. Finally, the paper explores some potential design methods for broadband and multi-band OPAs, discussing emerging technologies and highlighting potential research directions.