  Given a locally presentable enriched category $\mathcal{E}$ together with a
small dense full subcategory $\mathcal A$ of arities, we study the relationship
between monads on $\mathcal E$ and identity-on-objects functors out of
$\mathcal A$, which we call $\mathcal A$-pretheories. We show that the natural
constructions relating these two kinds of structure form an adjoint pair. The
fixpoints of the adjunction are characterised as the $\mathcal A$-nervous
monads---those for which the conclusions of Weber's nerve theorem hold---and
the $\mathcal A$-theories, which we introduce here.
  The resulting equivalence between $\mathcal A$-nervous monads and $\mathcal
A$-theories is best possible in a precise sense, and extends almost all
previously known monad--theory correspondences. It also establishes some
completely new correspondences, including one which captures the globular
theories defining Grothendieck weak $\omega$-groupoids.
  Besides establishing our general correspondence and illustrating its reach,
we study good properties of $\mathcal A$-nervous monads and $\mathcal
A$-theories that allow us to recognise and construct them with ease. We also
compare them with the monads with arities and theories with arities introduced
and studied by Berger, Melli\`es and Weber.
