In this chapter we define relaxed controls and the relaxed control problem and determine some of the properties of relaxed controls. For problems with well-behaved compact constraint sets, relaxed controls have a very useful compactness property. Also, at a given point in the subset R of (t, x) space, the set of directions that the state of a relaxed system may take is convex. This property is needed in existence theorems. We also shall prove an implicit function theorem for measurable functions that permits a definition of relaxed controls alternative to the one given in the next section. This theorem will also be used in our existence theorems. To motivate the definition of relaxed controls, we present two examples.