We study a nonlinear geometric theory of electromagnetism in this dissertation. In Part I, we consider the nonlinear Klein-Gordon equation coupled with the Born-Infeld equations under the electrostatic solitary wave ansatz. The existence of the least-action solitary waves is proved in both bounded smooth domain case and [special characters omitted] case. Especially, in bounded smooth domain case, we study the asymptotic behaviors and profiles of the positive least-action solitary waves with respect to the frequency parameter ω. We show that when κ and ω are suitably large, the positive least-action solitary waves admit only one local maximum point. When ω → ∞, the point-condensation phenomenon occurs if we consider the normalized least-action solitary waves. Compared to the U(1) model in Part I, we study a new static, SU(2) Born-Infeld-Higgs theory in the second part. The theory is constructed on the punctured space [special characters omitted]. The configuration space [special characters omitted] on which the SU(2) Born-Infeld-Higgs functional is finite is divided into infinitely many components. Each component is labeled by a monopole number vector in [special characters omitted]. A subspace [special characters omitted] on which the boundary values of |Φ| are non-zero admits a natural topology. By using this topology, we construct a one-one correspondence between the fundamental groups π₀(Map(S², S²))^N+1 and π₀([special characters omitted]).|