  We study conformal conserved currents in arbitrary irreducible
representations of the Lorentz group using the embedding space formalism. With
the help of the operator product expansion, we first show that conservation
conditions can be fully investigated by considering only two- and three-point
correlation functions. We then find an explicitly conformally-covariant
differential operator in embedding space that implements conservation based on
the standard position space operator product expansion differential operator
$\partial_\mu$, although the latter does not uplift to embedding space
covariantly. The differential operator in embedding space that imposes
conservation is the same differential operator $\mathcal{D}_{ijA}$ used in the
operator product expansion in embedding space. We provide several examples
including conserved currents in irreducible representations that are not
symmetric and traceless. With an eye on four-point conformal bootstrap
equations for four conserved vector currents $\langle JJJJ\rangle$ and four
energy-momentum tensors $\langle TTTT\rangle$, we mostly focus on conservation
conditions for $\langle JJ\mathcal{O}\rangle$ and $\langle
TT\mathcal{O}\rangle$. Finally, we reproduce and extend the consequences of
conformal Ward identities at coincident points by determining three-point
coefficients in terms of charges.
