In this dissertation, we provide a unified framework for analyzing immersed finite element methods in one spatial dimension, and we design a new geometry conforming IFE space in two dimensions with optimal approximation capabilities, alongside with applications to the elliptic interface problem and the hyperbolic interface problem. In the first part, we discuss a general m-th degree IFE space for one dimensional interface problems with many polynomial-like properties, then we develop a general framework for obtaining error estimates for the IFE spaces developed for solving a variety of interface problems, including but not limited to, the elliptic interface problem, the Euler-Bernoulli beam interface problem, the parabolic interface problem, the transport interface problem, and the acoustic interface problem. In the second part, we develop a new m-th degree finite element space based on the differential geometry of the interface to solve interface problems in two spatial dimensions. The proposed IFE space has optimal approximation capabilities, easy to construct, and the IFE functions satisfy the interface conditions exactly. We provide several numerical examples to demonstrate that the IFE space yields optimally converging solutions when applied to the elliptic interface problem and the hyperbolic interface problem with a symmetric interior penalty discontinuous Galerkin formulation. / Doctor of Philosophy / Interface problems appear naturally in many physics and Engineering applications where a physical quantity is considered across materials of different physical properties, such as heat transfer or sound propagation through different materials. Typically, these physical phenomena are modelled by partial differential equations with discontinuous coefficients representing the material properties. The main topics of this dissertation are about the development and analysis of immersed finite element methods for interface problems. The IFE method can use interface independent meshes, and employs approximating functions that capture the features of the solution at the interface. Specifically, we provide a unified framework for analyzing one-dimensional IFE problems, and we design a new framework to construct geometry conforming IFE spaces in two dimensions, with applications to the elliptic interface problem and the hyperbolic interface problem.