During the last years, great progress was made in material science in terms of concept, design and fabrication of new composite materials with conferred properties and desired functionalities. The scientific community paid particular interest to active soft materials, such as soft actuators, for their potential as transducers responding to various stimuli aiming to produce mechanical work. Inspired by this, materials engineers today are developing multidisciplinary approaches to produce new active matters, focusing on the kinematics allowed by the material itself more than on the possibilities offered by its design. Traditionally, more complex motions beyond pure elongation and bending are addressed by the robotics community. The present review targets encompassing and rationalizing a framework which will help a wider scientific audience to understand, sort and design future soft actuators and methods enabling complex motions. Special attention is devoted to recent progress in developing innovative stimulus-responsive materials and approaches for complex motion programming for soft robotics. In this context, a challenging overview of the new materials as well as their classification and comparison (performances and characteristics) are proposed. In addition, the great potential of soft transducers are outlined in terms of kinematic capabilities, illustrated by the related application. Guidelines are provided to design actuators and to integrate asymmetry enabling motions along any of the six basic degrees of freedom (translations and rotations), and strategies towards the programming of more complex motions are discussed. As a final note, a series of manufacturing methods are described and compared, from molding to 3D and 4D printing. The review ends with a Perspectives section, from material science and microrobotic points of view, on the soft materials’ future and close future challenges to be overcome.