  We introduce higher-order Massey products for algebras over algebraic
operads. This extends the work of Fernando Muro on secondary ones. We study
their basic properties and behavior with respect to morphisms of algebras and
operads and give some connections to formality. We prove that these
higher-order operations represent the differentials in a naturally associated
operadic Eilenberg--Moore spectral sequence. We also study the interplay
between particular choices of higher-order Massey products and quasi-isomorphic
$\mathcal P_\infty$-structures on the homology of a $\mathcal P$-algebra. We
focus on Koszul operads over a characteristic zero field and explain how our
results generalize to the non-Koszul case.
