Unitons, i.e.\ harmonic spheres in a unitary group, correspond to \lq uniton
bundles\rq, i.e.\ holomorphic bundles over the compactified tangent space to
the complex line with certain triviality and other properties. In this paper,
we use a monad representation similar to Donaldson's representation of
instanton bundles to obtain a simple formula for the unitons. Using the monads,
we show that real triviality for uniton bundles is automatic. We interpret the
uniton number as the `length' of a jumping line of the bundle, and identify the
uniton bundles which correspond to based maps into Grassmannians. We also show
that energy-$3$ unitons are $1$-unitons, and give some examples.