In this talk we report recent results and work in progress (joint with Tony Bloch) on various aspects of double-bracket flows and their generalisations. We present a computational algorithm, based on Magnus-type expansions of the underlying Lie algebra action and discuss different geometric and dynamical aspects of these flows. In particular we show that for the classical double-bracket flow $Y'=[[N,Y],Y]$, where (without loss of generality) $N$ is symmetric,

the distance $\|Y-N\|$ is minimised along the isospectral orbit by the stable fixed point in all $p$-Schatten norms for $p>1$ (Brockett alredy proved this for $p=2$, i.e. the Frobenius norm), as well as classifying all symmetric gauges for which fixed points are optimal (in the above sense) for $2\times 2$ matrices. We also consider generalised flows of the form $Y'=[Y,[g(Y-N),Y]]$ and discuss their (nontrivial) dynamics.