The RF (radio frequency) and microwave filter is widely used in various applications in the fields of radio broadcasting, radar, telecommunication and satellite technologies. The design methods of RF and microwave filters are important topics, especially for electromagnetic (EM) based design. In this thesis, a novel design method for sequential resonator-coupled bandpass filter is proposed by implementing the reflected group delay design approach with space mapping techniques. The theory of the reflected group delay method is discussed in detail. An improvement to the traditional reflected group delay method is proposed in which reflected group delay values at selected sweep frequency points are exploited as the target goals for each design stage instead of the whole curve used in the traditional method. Several filter design examples are given to verify the efficiency and accuracy of the improved reflected group delay method. An EM based design method is first proposed by implementing the aggressive space mapping technique as the optimization algorithm for the improved reflected group delay method in designing a 5-pole microstrip hairpin filter. The optimization routine using the reflected group delay method is represented mathematically and a design procedure is proposed for the integration of the aggressive space mapping technique and the reflected group delay method. The design steps are summarized and the filter is fabricated and tested. Another method to integrate the reflected group delay design approach with the implicit space mapping technique is also proposed. This method is applied to the design of a 6-pole microstrip hairpin filter. Detailed design theory and procedures are given. The 6-pole microstrip hairpin filter is designed using the Sonnet EM simulator. By using the proposed methods, the computation time and space mapping iterations are significantly reduced. The proposed methods are proven to be very efficient and accurate for EM-based sequential coupled resonator filter design compared to traditional EM based filter design methods.