  The heat kernel in curved space-time is computed to fourth order in a strict
expansion in the number of covariant derivatives. The computation is made for
arbitrary non abelian gauge and scalar fields and for the Riemann connection in
the coordinate sector. The expressions obtained hold for arbitrary tensor
representations of the matter field. Complete results are presented for the
diagonal matrix elements and for the trace of the heat kernel operator. In
addition, Chan's formula is extended to curved space-time. As a byproduct, the
bosonic effective action is also obtained to fourth order.
