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In this thesis we study the inverse Galois problem for del Pezzo surfaces of degree one and for algebraic K3 surfaces. We begin with an overview of how the question of the existence of $k$-points on a nice $k$-variety leads, via Brauer groups, to the inverse Galois problem. We then discuss an algorithm to compute all finite subgroups of the general linear group $\generalLinear(n,\integers)$ up to conjugacy. The first cohomologies of these subgroups contain all possible target groups of the inverse Galois problem for any family of nice $k$-varieties whose geometric Picard group is free and of finite rank. We apply these results to algebraic K3 surfaces defined over the rational numbers, providing explicit equations for a surface solving the only nontrivial instance of the inverse Galois problem in geometric Picard rank two. We also study representatives from three types of del Pezzo surfaces of degree one, searching for $5$-torsion in the Brauer group. For two of the three surfaces, we show that the Brauer group modulo constants is trivial when the surface is defined over the rational numbers, but becomes isomorphic to $\integers/5 \integers$ or $(\integers/5 \integers)^{2}$ when the base field is raised to a suitable number field. For the third surface, we show that its splitting field has degree 2400 as an extension of the rational numbers, a degree consistent with $5$-torsion in the Brauer group modulo constants.