A conflict-avoiding code (CAC) is a deterministic transmission scheme for asynchronous multiple access without feedback. When the number of simultaneously active users is less than or equal to $w$, a CAC of length $L$ with weight $w$ can provide a hard guarantee that each active user has at least one successful transmission within every consecutive $L$ slots. To deal with different individual performance requirements in heterogeneous systems, in this paper, we relax the identical-weight constraint in prior studies to study mixed-weight CACs for the first time. We first derive a new class of optimal CACs with constant weights, and then propose a general construction of mixed-weight CACs consisting of three different weights. Finally, we obtain a class of optimal mixed-weight CACs containing two different weights by the help of Kneser's Theorem and some techniques in Additive Combinatorics.