For any subset $A$ of a commutative ring $R$ (or, more generally, an $R$-module $M$) and any elements $\lambda_1, \dots, \lambda_k$ of $R$, let

\[\lambda_1 \cdot A + \cdots + \lambda_k \cdot A = \{\lambda_1 a_1 + \cdots + \lambda_k a_k : a_1, \dots, a_k \in A\}.\]

Such sums of dilates have attracted considerable attention in recent years, with the basic problem asking for an estimate on the minimum size of $|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A|$ given $|A|$. In this talk, I will discuss various generalizations and settings of this problem, and share recent progress. This is based on joint work with David Conlon.