Muon capture on the deuteron is a weak, two-nucleon process which can be related in a model-independent way to other experimentally inaccessible processes of astrophysical interest, such as solar p-p fusion and neutrino-deuteron scattering. The MuSun experiment measures the disappearance rate of negative muons stopped in an ultra-pure deuterium target optimized to prepare mu-d atoms in the doublet hyperfine state. Muon decay times are measured, and the resulting muon lifetime in deuterium is compared to the known positive muon lifetime to determine the doublet capture rate with a goal of 1.5% precision. The target operates as a low-noise, high resolution time projection chamber (TPC), providing highly stringent event selection by constructing muon tracks within the target. The capture rate, being approximately 1000 times slower than free muon decay, demands a ten part-per-million measurement of the disappearance rate, and thus of order ten billion candidate events with careful event selection and control of systematic effects. We have acquired approximately 12 billion candidate events at the piE1 muon beam line at the Paul Scherrer Institute over two experimental campaigns in 2014 and 2015. The experiment offers a unique opportunity to probe the two-body weak current, and presents a number of unique challenges to achieve the desired precision such as high chemical and isotopic target purity, muon beam related backgrounds, and the effect of muon-catalyzed fusion events. Here we will present the experimental and theoretical context for the measurement, give an overview of the experiment, and focus on recent challenges and solutions which will lead to a determination of the capture rate with unprecedented precision.