Welding of single crystal nickel-based superalloys leads to excessive stray grain formation as well as cracking associated with stray grain boundaries. This paper examines the microstructure development in both model alloys and commercial alloys and progress in terms of modeling the dendritic growth pattern produced in these alloys. The tendency to form stray grains is also studied, and recent experimental work that supports the principal of constitutional supercooling as the underlying mechanism for stray grain formation is presented.