Grasping and manipulation are challenging tasks that are nonetheless critical for many robotic systems and applications. A century ago, robots were conceived as humanoid automata. While conceptual at the time, this viewpoint remains influential today. Many robotic grippers have been inspired by the dexterity and functionality of the prehensile human hand. However, multi-fingered grippers that emulate the hand often integrate many kinematic degrees-of-freedom, and thus complex mechanisms, which must be controlled in order to grasp and manipulate objects. Soft fingers can facilitate grasping through intrinsic compliance, enabling them to conform to diverse objects. However, as with conventional fingered grippers, grasping via soft fingers involves challenges in perception, computation, and control, because fingers must be placed so as to achieve force closure, which depends on the shape and pose of the object. Emerging soft robotics research on non-anthropomorphic grippers has yielded new techniques that can circumvent fundamental challenges associated with grasping via fingered grippers. Common to many non-anthropomorphic soft grippers are mechanisms for morphological deformation or adhesion that simplify the grasping of diverse objects in different poses, without detailed knowledge of the object geometry. These advantages may allow robots to be used in challenging applications, such as logistics or rapid manufacturing, with lower cost and complexity. In this perspective, we examine challenges associated with grasping via anthropomorphic grippers. We describe emerging soft, non-anthropomorphic grasping methods, and how they may reduce grasping complexities. We conclude by proposing several research directions that could expand the capabilities of robotic systems utilizing non-anthropomorphic grippers.