Over the last two decades, smart materials have drawn significant attention and growing interest in a broad range of engineering applications because of their unique engineering features. In this article, robotic applications using a class of smart materials that have the ability to actuate when subjected to external stimuli, such as magnetic or electric fields, temperature and voltage are comprehensively reviewed. Smart materials used for robotic applications include electro-rheological fluids, magnetorheological fluids, shape memory alloys, and piezoelectric transducers. This article describes the attributes and inherent properties of individual smart materials that make them suitable for actuating robotic applications and discusses their associated technical limitations, in order to address the design challenges. In addition, many types of robots proposed to date, such as flexible robot arm, and biomimetic and soft robots are introduced in terms of their dynamic modeling, design, controlling, and practical applications. This article provides an updated review of recent smart material research for robotic applications, with over a hundred state-of-the-art references categorized into various types of smart materials.