This dissertation studies alternate discriminants, a class of invariants for number fields based on the standard discriminant, and their mass formulas. We study alternate discriminants both in their combinatorial properties and in their relationships to field-counting heuristics of Malle and Bhargava. Chapter 1 is an exposition of existing results and conjectures on counting number fields by standard discriminant. In Chapter 2, we define weighted discriminants, the objects of primary interest in the remainder of this work, as well as other relevant ideas. We then prove our main theorem, which restricts the number of counting functions of a certain type for any given group that can have mass formulas. In Chapter 3, we extend the techniques and machinery of Chatper 2 to analyze questions about class groups of number fields. Where possible, we compare our predicted asymptotics for class groups to known results and to conjectures of Cohen-Lenstra and Cohen-Martinet. In Chapter 4, we calculate mass formulas, or prove that none exist, for number fields with several specific Galois groups. Finally, Chapter 5 discusses some further questions that we leave open for future work.