In this talk, I will show how techniques from Combinatorial Optimization and Combinatorics can be leveraged to achieve progress on a well-known question in Integer Programming, namely how to solve integer linear programs (ILPs) with bounded subdeterminants. More precisely, I will present an efficient (even strongly polynomial) algorithm to solve ILPs defined by a constraint matrix whose subdeterminants are all within {-2,-1,0,1,2}. This problem class naturally extends the well-known class of ILPs with a totally unimodular (TU) constraint matrix, i.e., where subdeterminants are within {-1,0,1}, and captures some open problems. We employ a variety of techniques common in Combinatorial Optimization, including polyhedral techniques, matroids, and submodular functions. In particular, using a polyhedral result of Veselov and Chirkov, we reduce the problem to a combinatorial one with a so-called parity constraint. We then show how a seminal, purely combinatorial result of Seymour on decomposing TU matrices, which is tightly related to regular matroids and was originally developed in this context, can be used algorithmically to break the problem into smaller ones. Finally, these smaller problems are amenable to combinatorial optimization techniques like parity-constrained submodular minimization. Additionally, I will highlight some of the many open problems in this field and discuss recent related results.