This paper presents a new screw theory that deals with “lines” and “line systems”. A line is a screw with a zero pitch. A rank m screw system is called a line screw system if it contains m linearly independent lines. The set of lines within a screw system is called the line variety. A general m-system is a line screw system if the rank of its line variety equals m. This paper will answer two questions: (1) how to calculate the rank of the line variety for a given m-system and (2) how to algorithmically find a set of linearly independent lines from the given screw system. The motivation of this work comes from flexure synthesis whose goal is to find a pattern of flexures to achieve a specified motion. It has been previously found that a wire flexure is considered a line screw, or more specifically a pure force wrench. The synthesis of flexures is to remove undesired motion by applying constraints with flexures. By following reciprocity and definitions of line screws, we have derived the sufficient and necessary conditions of line screw systems. When applied to flexure synthesis, we show that not all motion pattern can be realized with wire flexures connected in parallel. A computational algorithm based on the line screw theory is developed to find all admissible line screws or force wrenches for a given motion space. At last two flexure synthesis case studies are provided to demonstrate the theory and the algorithm.Copyright © 2010 by ASME