Pedestrian navigation activity recognition (PNAR) has a significant impact on positioning and tracking performance. Smartphone-based PNAR utilizes measurements from sensors embedded in smartphone to identify pedestrian motion mode and smartphone usage mode. Compared with other PNAR technologies, smartphone-based PNAR has the advantage of autonomy and practicality. Though various PNAR recognition methods based on smartphone have been proposed, up-to-date review papers that summarize relevant technologies, methods, and solutions of PNAR are relatively less so far in the literature. This work aims to present an elaborated, timely, and valuable survey for different smartphone-based PNAR techniques, including sensor calibration, pedestrian modeling, pedestrian motion and smartphone usage modes definition, workflow, state-of-the-art research methods, public datasets, performance evaluation metrics, and relative competitions. Finally, applications, current challenges and future research trends are discussed. The work propels a better understanding of existing navigation activity recognition methods. It is helpful for researchers to further design PNAR methods, and develop more accurate and robust pedestrian positioning and navigation systems.