A reciprocity formula usually relates certain moments of two different families of $L$-functions which apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central $L$-values for $\mathrm{GL}(2)$. In this talk, we describe some instances of reciprocity formulas both in low and high-rank groups and give certain applications to subconvexity and non-vanishing of central $L$-values. These are based on the joint works with Nunes (https://arxiv.org/abs/2111.02297) and Blomer–Nelson (https://arxiv.org/abs/2404.10692).