The adaptability of natural organisms in altering body shapes in response to the environment has inspired the development of artificial morphing matter. These materials encode the ability to transform their geometrical configurations in response to specific stimuli and have diverse applications in soft robotics, wearable electronics, and biomedical devices. However, achieving the morphing of intricate three-dimensional shapes from a two-dimensional flat state is challenging, as it requires manipulations of surface curvature in a controlled manner. In this review, we first summarize the mechanical principles extensively explored for realizing morphing matter, both at the material and structural levels. We then highlight its applications in the soft robotics field. Moreover, we offer insights into the open challenges and opportunities that this rapidly growing field faces. This review aims to inspire researchers to uncover innovative working principles and create multifunctional morphing matter for various engineering fields.