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1. Every year billions of insects engage in long‐distance, seasonal mass migrations which have major consequences for agriculture, ecosystem services and insect‐vectored diseases. Tracking this movement in the field is difficult, with mass migrations often occurring at high altitudes and over large spatial scales. 2. As such, tethered flight provides a valuable tool for studying the flight behaviour of insects, giving insights into flight propensity (e.g. distance, duration and velocity) and orientation under controlled laboratory settings. By experimentally manipulating a variety of environmental and physiological traits, numerous studies have used this technology to study the flight behaviour of migratory insects ranging in size from aphids to butterflies. Advances in functional genomics promise to extend this to the identification of genetic factors associated with flight. Tethered flight techniques have been used to study migratory flight characteristics in insects for more than 50 years, but have never been reviewed. 3. This study summarises the key findings of this technology, which has been employed in studies of species from six Orders. By providing detailed descriptions of the tethered flight systems, the present study also aims to further the understanding of how tethered flight studies support field observations, the situations under which the technology is useful and how it might be used in future studies. 4. The aim is to contextualise the available tethered flight studies within the broader knowledge of insect migration and to describe the significant contribution these systems have made to the literature.

Let $t$ and $k$ be a positive integers, and let $I_k=\{i\in \mathbb{Z}:\; -k\leq i\leq k\}$. Let $\mathsf{s}'_t(I_k)$ be the smallest positive integer $\ell$ such that every zero-sum sequence $S$ over $I_k$ of length $|S|\ge \ell$ contains a zero-sum subsequence of length $t$. If no such $\ell$ exists, then let $\mathsf{s}'_t(I_k)=\infty$. In this paper, we prove that $\mathsf{s}'_t(I_k)$ is finite if and only if every integer in $[1,D(I_k)]$ divides $t$, where $D(I_k)=\max\{2,2k-1\}$ is the Davenport constant of $I_k$. Moreover, we prove that if $\mathsf{s}'_t(I_k)$ is finite, then $t+k(k-1)\leq \mathsf{s}'_t(I_k)\leq t+(2k-2)(2k-3)$. We also show that $\mathsf{s}'_t(I_k)=t+k(k-1)$ holds for $k\leq 3$ and conjecture that this equality holds for any $k\geq1$., Comment: 13 pages. Added a new reference, corrected typos in the proof of Lemma 13, and added more details to the proof of Theorem 5