This paper discusses the design and analysis of machining fixtures from a kinematics perspective. We use screw theory to model the fixture elements, the motions and forces/moments involved. We present linear algebraic methods that are quick and reasonably accurate, as opposed to existing methods. First, we study the determinism of 3-2-1 locator schemes and present simple formal geometric conditions to verify it. They also give an insight into the design issues influencing the robustness of the locator scheme. Next, we discuss the problem of synthesizing optimal clamping schemes that minimize the maximum clamping force. We present linear programs to solve this problem with and without friction. Finally, we focus our attention on the robustness of the fixturing scheme and discuss the use of form closure to provide it. We present linear algebraic methods to compute the regions of the workpart within which the clamps can be placed to produce form closure, both dependently and independently of each other. The method is easily incorporated in the linear program to compute optimal clamping schemes.