Joints allow relative motions between the rigid structural elements (termed ‘links’) they connect. As joints are crucial for many animal movements used in locomotion, feeding, and reproduction, an understanding of the biomechanical principles of joint form and function is essential. There are two categories of animal joints: sliding and flexible. Sliding joints (e.g. vertebrate and arthropod articulated joints) transmit compressional forces directly through contact between the links. The shapes of the contact surfaces and their connective tissue capsules limit their possible motions. Flexible joints include pliable connections which allow relative motions between the links. They are normally loaded in tension because compressional forces tend to cause the connection to buckle. As flexible joints are less well-understood, they are the focus of this dissertation. In particular, I define, describe, and analyze a new form of flexible joint: the muscle articulation. In muscle articulations, the pliable connections consist of multiple orientations of muscle fibers or myofibrils arranged as muscular hydrostats. Arrangements of connective tissue fibers are also important and may limit shape change and store elastic energy. Together, these soft tissues allow the multi-functional characteristics of muscle articulations; they connect links, generate motive forces, create pivot areas, and transmit compressional forces between the links. These arrangements, with appropriate neural control, may allow muscle articulations to produce a relatively great diversity and complexity of movements. Three examples of muscle articulations were examined. First, I described the morphology of the octopus buccal mass in detail. A biomechanical analysis of this morphology generated hypotheses of function that were tested experimentally by correlating muscle activation patterns with beak movement. Second, the morphology and biting movements of the eversible proboscidial jaws of the polychaete Nereis were analyzed and found to share key muscle articulation soft tissue characteristics. Third, the morphology of the armed proboscides of the kalyptorhynchs, a group of meiofaunal turbellarian flatworms, was analyzed and identified as a unique muscle articulation of simple and microscopic construction. The comparative analyses of these joints not only identified common muscle articulation characteristics, but demonstrate that this important type of joint has evolved independently in at least three invertebrate phyla.