Birational superrigidity and K-stability are properties of Fano varieties that have many interesting geometric implications. For instance, birational superrigidity implies non-rationality and K-stability is related to the existence of Kähler-Einstein metrics. Nonetheless, both properties are hard to verify in general. In this talk, I will first explain how to relate birational superrigidity to K-stability using alpha invariants; I will then outline a method of proving birational superrigidity that works quite well with most families of index one Fano complete intersections and thereby also verify their K-stability. This is partly based on joint work with Charlie Stibitz and Yuchen Liu.