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We study the existence and left properness of transferred model structuresfor "monoid-like" objects in monoidal model categories. These include genuinemonoids, but also all kinds of operads as for instance symmetric, cyclic,modular, higher operads, properads and PROP's. All these structures can berealised as algebras over polynomial monads.  We give a general condition for a polynomial monad which ensures theexistence and (relative) left properness of a transferred model structure forits algebras. This condition is of a combinatorial nature and singles out aspecial class of polynomial monads which we call tame polynomial. Manyimportant monads are shown to be tame polynomial.

Homotopy theory for algebras over polynomial monads