  We prove some results for the solitons of the Ricci-Bourguignon flow,
generalizing corresponding results for Ricci solitons. Taking motivation from
Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon
$almost$ solitons and prove some results about them which generalize previous
results for Ricci almost solitons. We also derive integral formulas for compact
gradient Ricci-Bourguignon solitons and compact gradient Ricci-Bourguignon
almost solitons. Finally, using the integral formula we show that a compact
gradient Ricci-Bourguignon almost soliton is isometric to an Euclidean sphere
if it has constant scalar curvature or its associated vector field is
conformal.
