The field of computer vision studies the computational tools and methods required for computers to be able to process visual information, for example images and video. Shape descriptors are one of the tools commonly used in image processing applications. Shape descriptors are mathematical functions which are applied to an image and produce numerical values which are representative of a particular characteristic of the image. These numerical values can then be processed in order to provide some information about the image. For example, these values can be fed to a classifier in order to assign a class label to the image. There are a number of shape descriptors already existing in the literature for 2D and 3D images. The aim of this thesis is to develop additional shape descriptors which provide an improvement over (or an alternative to) those already existing in the literature. A large majority of the existing 2D shape descriptors use surface information to produce a measure. However, in some applications surface information is not present and only partially extracted contours are available. In such cases, boundary based shape descriptors must be used. A new boundary based shape descriptor called Linearity is introduced. This measure can be applied to open or closed curve segments. In general the availability of 3D images is comparatively smaller than that of 2D images. As a consequence, the number of existing 3D shape descriptors is also relatively smaller. However, there is an increasing interest in the development of 3D descriptors. In this thesis we present two basic 3D measures which afterwards are modified to produce a range of new shape descriptors. All of these descriptors are similar in their behaviour, however they can be combined and applied in different image processing applications such as image retrieval and classification. This simple fact is demonstrated through several examples.