Tactile sensing is an essential element of autonomous dexterous robot hand manipulation. It provides information about forces of interaction and surface properties at points of contact between the robot fingers and the objects. Recent advancements in robot tactile sensing led to development of many computational techniques that exploit this important sensory channel. This paper reviews current state-of-the-art of manipulation and grasping applications that involve artificial sense of touch and discusses pros and cons of each technique. The main issues of artificial tactile sensing are addressed. General requirements of a tactile sensor are briefly discussed and the main transduction technologies are analyzed. Twenty eight various tactile sensors, each integrated into a robot hand, are classified in accordance with their transduction types and applications. Previously issued reviews are focused on hardware part of tactile sensors, whereas we present an overview of algorithms and tactile feedback-based control systems that exploit signals from the sensors. The applications of these algorithms include grasp stability estimation, tactile object recognition, tactile servoing and force control. Drawing from advancements in tactile sensing technology and taking into consideration its drawbacks, this paper outlines possible new directions of research in dexterous manipulation.