  Under a minimum of assumptions, we develop in generality the basic theory of
universal algebra in a symmetric monoidal closed category $\mathcal{V}$ with
respect to a specified system of arities $j:\mathcal{J} \hookrightarrow
\mathcal{V}$. Lawvere's notion of algebraic theory generalizes to this context,
resulting in the notion of single-sorted $\mathcal{V}$-enriched
$\mathcal{J}$-cotensor theory, or $\mathcal{J}$-theory for short. For suitable
choices of $\mathcal{V}$ and $\mathcal{J}$, such $\mathcal{J}$-theories include
the enriched algebraic theories of Borceux and Day, the enriched Lawvere
theories of Power, the equational theories of Linton's 1965 work, and the
$\mathcal{V}$-theories of Dubuc, which are recovered by taking $\mathcal{J} =
\mathcal{V}$ and correspond to arbitrary $\mathcal{V}$-monads on $\mathcal{V}$.
We identify a modest condition on $j$ that entails that the
$\mathcal{V}$-category of $\mathcal{T}$-algebras exists and is monadic over
$\mathcal{V}$ for every $\mathcal{J}$-theory $\mathcal{T}$, even when
$\mathcal{T}$ is not small and $\mathcal{V}$ is neither complete nor
cocomplete. We show that $j$ satisfies this condition if and only if $j$
presents $\mathcal{V}$ as a free cocompletion of $\mathcal{J}$ with respect to
the weights for left Kan extensions along $j$, and so we call such systems of
arities eleutheric. We show that $\mathcal{J}$-theories for an eleutheric
system may be equivalently described as (i) monads in a certain one-object
bicategory of profunctors on $\mathcal{J}$, and (ii) $\mathcal{V}$-monads on
$\mathcal{V}$ satisfying a certain condition. We prove a characterization
theorem for the categories of algebras of $\mathcal{J}$-theories, considered as
$\mathcal{V}$-categories $\mathcal{A}$ equipped with a specified
$\mathcal{V}$-functor $\mathcal{A} \rightarrow \mathcal{V}$.
