The high glass-formation tendency of Pd-Si-Cu alloys is well established, particularly in the compositional vicinity of Pd77.5Si16.5Cu6, and non-crystalline solids have been formed in Pd-Si-Cu liquids, using a number of rapid cooling techniques. Of course, understanding the competition between the solidification and the relaxation to a non-crystalline arrested or glassy state requires elucidation of the phases and solidification paths available at high undercooling. In the present work, directional growth in Pd77.5Si16.5Cu6 is investigated using velocities of 0.005 to 7 mm/s. A detailed thermodynamic model for the Pd-Si-Cu system is developed to examine solidification pathway selection, comparing the competing single-phase and multiphase (invariant and univariant) modes, with structures extending down to the nanometer scale before further increases in velocity give rise to the glassy state. Work at Ames Laboratory was supported by the US Department of Energy, Basic Energy Sciences, Division of Materials Science and Engineering, under Contract No. DE-AC02-07CH11358.