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Soft and continuum robots are transforming medical interventions thanks to their flexibility, miniaturization, and multidirectional movement abilities. Although flexibility enables reaching targets in unstructured and dynamic environments, it also creates challenges for control, especially due to interactions with the anatomy. Thus, in recent years lots of efforts have been devoted for the development of shape reconstruction methods, with the advancement of different kinematic models, sensors, and imaging techniques. These methods can increase the performance of the control action as well as provide the tip position of robotic manipulators relative to the anatomy. Each method, however, has its advantages and disadvantages and can be worthwhile in different situations. For example, electromagnetic (EM) and Fiber Bragg Grating (FBG) sensor-based shape reconstruction methods can be used in small-scale robots due to their advantages thanks to miniaturization, fast response, and high sensitivity. Yet, the problem of electromagnetic interference in the case of EM sensors, and poor response to high strains in the case of FBG sensors need to be considered. To help the reader make a suitable choice, this paper presents a review of recent progress on shape reconstruction methods, based on a systematic literature search, excluding pure kinematic models. Methods are classified into two categories. First, sensor-based techniques are presented that discuss the use of various sensors such as FBG, EM, and passive stretchable sensors for reconstructing the shape of the robots. Second, imaging-based methods are discussed that utilize images from different imaging systems such as fluoroscopy, endoscopy cameras, and ultrasound for the shape reconstruction process. The applicability, benefits, and limitations of each method are discussed. Finally, the paper draws some future promising directions for the enhancement of the shape reconstruction methods by discussing open questions and alternative methods.