The Tactile Internet will enable users to physically explore remote environments and to make their skills available across distances. An important technological aspect in this context is the acquisition, compression, transmission, and display of haptic information. In this paper, we present the fundamentals and state of the art in haptic codec design for the Tactile Internet. The discussion covers both kinesthetic data reduction and tactile signal compression approaches. We put a special focus on how limitations of the human haptic perception system can be exploited for efficient perceptual coding of kinesthetic and tactile information. Further aspects addressed in this paper are the multiplexing of audio and video with haptic information and the quality evaluation of haptic communication solutions. Finally, we describe the current status of the ongoing IEEE standardization activity P1918.1.1 which has the ambition to standardize the first set of codecs for kinesthetic and tactile information exchange across communication networks.