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In this thesis, we study six Ginzburg-Landau minimization problems in the context of two-dimensional nematic liquid crystals with the intention of finding conditions for the existence of boundary vortices. The first minimization problem consists of the standard Ginzburg-Landau energy on bounded, simply connected domains Ω ⊂ R2 with boundary energy penalizing minimizers who stray from being parallel to some smooth S1-valued boundary function g of degree D ≥ 1. The second and third minimization problems consider the same Ginzburg-Landau energy but now with divergence and curl penalization in the interior and boundary function taken to be g = τ, the positively oriented unit tangent vector to the boundary. The remaining three problems involve minimizing the same energies, but now over the set for which all functions are precisely parallel to the given boundary data (up to a set for which their norms can be zero). These six problems are classified under two categories called the weak and strong orthogonal problems. In each of the six problems, we show that conditions exist for which sequences of minimizers converge to a limiting S1-valued vector field describing an equilibrium configuration for nematic material with defects. In some cases, energy estimates are obtained that show vortices belong to the boundary exclusively and the exact number of these vortices are known. A special case is also studied in the strong orthogonality setting. The analysis here suggests that geometries exist for which boundary vortices may be energetically preferable to interior vortices in the case where interior and boundary vortices have similar energy contributions. / Thesis / Doctor of Philosophy (PhD)