For a continuum, we consider the equivalence relation in which two points are equivalent if they can be joined by an arc. This equivalence relation is analytic in general (i.e. a continuous image of a Polish space). Recently, Debs and Saint Raymond proved that, for planar continua, this equivalence relation is always Borel measurable. We show that for every planar continuum, the arc-connection relation is in fact Borel reducible to the Vitali equivalence relation, where two real numbers are equivalent if their difference is rational. Moreover, the Knaster continuum is an example where this complexity is attained. This is joint work with Michal Hevessy and Yusuf Uyar. We also investigate several related questions concerning continuum-wise connectivity.