In this thesis, we mainly study the structure of $D_9$-brane Chan-Paton factors in Type II orientifold $O_p^{\pm}$ theories. An explicit structure is found through a thorough and systematic derivation. For this purpose, we give a complete analysis of the worldsheet parity square action for all p-q open strings (i.e, the strings starting from $D_p$-branes and ending on $D_q$-branes), which is an extension of Gimon-Polchinski's work on the parity square action for p-q strings with $p-q$ even. We also formulate the rules for computing the scattering amplitudes of open strings ending on D-branes. These are important to build phenomenological models in string theory. As an application to the mathematical aspects of string theory, we confirm the proposal that the D-brane charges in Type II orientifolds are classified by the KR-theory groups. All these results will appear in our paper\footnote{Dongfeng Gao and Kentaro Hori, Clifford Algebras and Chan-Paton Factors, to appear.}. We also show some preliminary results on the B-type D-branes in the orientifolds of linear sigma models. A typical linear sigma model has both a geometric phase and a Landau-Ginzberg phase over its K\"ahler moduli space. The B-type D-branes are described by suitable categories in a linear sigma model and its phases. The orientifold projection induces parity mappings on the various categories. By looking at the transportation of invariant branes in the moduli space, we find the relations among the parity mappings of different phases in the quintic model.

In this thesis, we mainly study the structure of $D_9$-brane Chan-Paton factors in Type II orientifold $O_p^{\pm}$ theories. An explicit structure is found through a thorough and systematic derivation. For this purpose, we give a complete analysis of the worldsheet parity square action for all p-q open strings (i.e, the strings starting from $D_p$-branes and ending on $D_q$-branes), which is an extension of Gimon-Polchinski's work on the parity square action for p-q strings with $p-q$ even. We also formulate the rules for computing the scattering amplitudes of open strings ending on D-branes. These are important to build phenomenological models in string theory. As an application to the mathematical aspects of string theory, we confirm the proposal that the D-brane charges in Type II orientifolds are classified by the KR-theory groups. All these results will appear in our paper\footnote{Dongfeng Gao and Kentaro Hori, Clifford Algebras and Chan-Paton Factors, to appear.}. We also show some preliminary results on the B-type D-branes in the orientifolds of linear sigma models. A typical linear sigma model has both a geometric phase and a Landau-Ginzberg phase over its K\"ahler moduli space. The B-type D-branes are described by suitable categories in a linear sigma model and its phases. The orientifold projection induces parity mappings on the various categories. By looking at the transportation of invariant branes in the moduli space, we find the relations among the parity mappings of different phases in the quintic model.