Nowadays, harvesting delicate and high-value fruits, vegetables and edible fungi requires a large input of manual human labor. The relatively low wages and many health problems the workforce faces make this profession increasingly unpopular. Meanwhile, robotic systems that selectively harvest crops are being developed. Whilst the moving platform, manipulator, and recognition systems of such robots are studied the past few decades, research on the gripping end of such robots is only recently growing. This study analyses the state-of-the-art of soft grippers for crop handling and harvesting, reporting their quantitative and qualitative characteristics. Seventy-eight grippers are retrieved from the academic literature and compared with each other in terms of their design and reported performance, more specifically grasping and detachment methods, materials used, type of actuators and sensors employed, and the control of the gripping procedure. In addition, the identified grippers are classified into 13 distinct soft grasping technology categories. Moreover, the retrieved papers are analyzed with respect to their publication date and country of origin to observe the recent growth in the field. Furthermore, a subset of soft grippers is identified that was tested on the task of selectively harvesting crops, where grip and detachment success rates and plant and crop damage are compared.