  We provide conditions on a monoidal model category $\mathcal{M}$ so that the
category of commutative monoids in $\mathcal{M}$ inherits a model structure
from $\mathcal{M}$ in which a map is a weak equivalence or fibration if and
only if it is so in $\mathcal{M}$. We then investigate properties of
cofibrations of commutative monoids, rectification between $E_\infty$-algebras
and commutative monoids, the relationship between commutative monoids and
monoidal Bousfield localization functors, when the category of commutative
monoids can be made left proper, and functoriality of the passage from a
commutative monoid $R$ to the category of commutative $R$-algebras. In the
final section we provide numerous examples of model categories satisfying our
hypotheses.
