Research on metamaterials has been growing ever since the first experimental realization of double negative media. The theory of transformation optics provides people with a perfect tool to make use of vast possibilities of the constitutive parameters for metamaterials. A lot of fascinating designs have been brought to us by transformation optics, with invisibility cloaks being the most intensely studied. The present thesis aims to develop the basic theory of transformation optics, and utilize it to design invisibility cloaks for various applications. After the background description of this field, the theory of transformation optics is first introduced. Formulas of transformation medium parameters and transformed fields are derived with every detail explained, so that the working knowledge of transformation optics can be grasped with minimal prerequisite mathematics. Proof of form invariance of full Maxwell’s equations with sources is presented. Design procedure of transformation optics is then demonstrated by creating perfect invisibility cloaks. The introduction to basic theory is followed by discussions on our works included in our published papers. As our first application, a method of designing two-dimensional reduced cloaks of complex shapes is proposed to relieve the difficulty of singularity occurring in perfect cloaks. The simple and intuitive method is the first way to design two-dimensional reduced cloaks of shapes other than cylindrical. Elliptical and bowtie shaped reduced cloaks are presented to verify the effectiveness of the method. Prominent scattering reduction is observed for both examples. Considering the practical realization, transformations continuous in the whole space must be the identity operation outside certain volume, and thus they can only manipulate fields locally. Discontinuous transformations are naturally considered to break the limitation. We study the possible reflections from such a transformation medium due to a discontinuous transformation by a new concept of inverse transformation. This way, the reflection falls into the framework of transformation optics as well. A necessary and sufficient condition for no reflection is derived as a special case. Unlike the invisibility realized by perfect cloaks, cloaking an object over a dielectric half-space has advantages in some particular applications. Starting from a perfect cloak, a half-space cloak is designed to achieve this. In our design, two matching strips embedded in the dielectric ground are used to induce proper reflection in the upper air space, so that the reflected field is the same as that from the bare dielectric ground. Cloaks obtained from singular transformations and even reduced models all have null principal value in their material parameters, making invisibility inherently very narrowband. In contrast, a carpet cloak designed by only coordinate deformation does not have the narrowband issue, and can perform well in a broad spectrum. The invisibility accomplished by the carpet cloak is also for the half-space case as our previous design. In this part, we extend the original version of a carpet cloak above a PEC sheet to a general dielectric ground. / QC 20110415