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. In this paper, we generalize some previously known constructions of constant-weight CACs, and then derive several classes of optimal CACs by the help of Kneser's Theorem and some techniques in Additive Combinatorics. Another spotlight of this paper is to relax the identical-weight constraint in prior studies to study mixed-weight CACs for the first time, for the purpose of increasing the throughput and reducing the access delay of some potential users with higher priority. As applications of those obtained optimal CACs, we derive some classes of optimal mixed-weight CACs.