Space-based infrastructure is rapidly expanding in the near-Earth and cislunar regimes, the region of space dominated by the gravity of the Earth and Moon. As more spacecraft operate in these regions, it becomes increasingly important to understand and predict their motion, particularly in the presence of uncertainty. In the Geometry-Informed Spaceflight, Mechanics, and Orbits (GISMO) Lab, my research focuses on Hamiltonian methods in astrodynamics and the practical applications of symplectic geometry to spacecraft autonomy, guidance, and navigation. Many spaceflight problems are modeled as Hamiltonian systems, which impose geometric constraints on how regions of phase space evolve. The GISMO lab leverages Hamiltonian mechanics, symplectic geometry, dynamical systems theory, and estimation theory to develop numerical and semi-analytical tools. These tools aim to improve spacecraft navigation, uncertainty quantification, transport analysis, and optimal control.