For each closed curve we can determine the width in any direction. If the width of a curve is the same in all directions, then it is called a curve of constant width. A common example of such a curve is a circle. Apart from the circle, there are infinitely many other curves of constant width. The simplest example of such curves is the curvilinear triangle, also called the Reuleaux triangle because it was discovered by German scientist and engineer Franz Reuleaux. In the first part of this graduate thesis we describe different ways of constructing curves of constant width. Curves of constant width have many interesting properties, and some of them are mentioned and proven in this work. For example, Barbier’s theorem says that a curve of constant width h and a circle of diameter h have the same length. This and other statements on length and the area enclosed by such curves are given and illustrated. The so called Approximation Theorem for curves of constant width is also proved. A curve of constant width is the boundary of the figure that we call a figure of constant width. The three-dimensional analogy of the figure of constant width is the body of constant width. According to the way they are obtained, they are divided into rotational and non-rotational bodies. Two types of non-rotational bodies of constant width were discovered by Swiss mathematician Ernst Meissner. A short exposition on bodies of constant width makes the last part of this thesis.