Consider the maximal function associated with averages over dilates of the helix (or, more generally, of any curve with non-vanishing curvature and torsion). This object can be seen as a 3-dimensional analogue of the classical circular maximal function in the plane, studied by Bourgain (sharp $L^p$ bounds for $p>2$) and Schlag and Schlag-Sogge (sharp $L^p$-$L^q$ bounds). In this talk, we report sharp 3-dimensional versions of those well-known 2-dimensional results, which use recent developments in multilinear harmonic analysis. This is based on joint works with Shaoming Guo, Jonathan Hickman and Andreas Seeger ($L^p$ bounds), and Jennifer Duncan and Jonathan Hickman ($L^p$-$L^q$ bounds).