  Quantitative algebras are $\Sigma$-algebras acting on metric spaces, where
operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic
varieties as categories of quantitative algebras presented by quantitative
equations. We prove that for the category $\mathsf{UMet}$ of ultrametric spaces
such varieties bijectively correspond to strongly finitary monads on
$\mathsf{UMet}$. The same holds for the category $\mathsf{Met}$ of metric
spaces, provided that strongly finitary endofunctors are closed under
composition.
  For uncountable cardinals $\lambda$ there is an analogous bijection between
varieties of $\lambda$-ary quantitative algebras and monads that are strongly
$\lambda$-accessible. Moreover, we present a bijective correspondence between
$\lambda$-basic varieties as introduced by Mardare et al and enriched,
surjections-preserving $\lambda$-accesible monads on $\mathsf{Met}$. Finally,
for general enriched $\lambda$-accessible monads on $\mathsf{Met}$ a bijective
correspondence to generalized varieties is presented.
