  For a Dirichlet series symbol $g(s) = \sum_{n \geq 1} b_n n^{-s}$, the
associated Volterra operator $\mathbf{T}_g$ acting on a Dirichlet series
$f(s)=\sum_{n\ge 1} a_n n^{-s}$ is defined by the integral $f\mapsto
-\int_{s}^{+\infty} f(w)g'(w)\,dw$. We show that $\mathbf{T}_g$ is a bounded
operator on the Hardy space $\mathcal{H}^p$ of Dirichlet series with $0 < p <
\infty$ if and only if the symbol $g$ satisfies a Carleson measure condition.
When appropriately restricted to one complex variable, our condition coincides
with the standard Carleson measure characterization of
${\operatorname{BMOA}}(\mathbb{D})$. A further analogy with classical
${\operatorname{BMO}}$ is that $\exp(c|g|)$ is integrable (on the infinite
polytorus) for some $c > 0$ whenever $\mathbf{T}_g$ is bounded. In particular,
such $g$ belong to $\mathcal{H}^p$ for every $p < \infty$. We relate the
boundedness of $\mathbf{T}_g$ to several other ${\operatorname{BMO}}$ type
spaces: ${\operatorname{BMOA}}$ in half-planes, the dual of $\mathcal{H}^1$,
and the space of symbols of bounded Hankel forms. Moreover, we study symbols
whose coefficients enjoy a multiplicative structure and obtain coefficient
estimates for $m$-homogeneous symbols as well as for general symbols. Finally,
we consider the action of $\mathbf{T}_g$ on reproducing kernels for appropriate
sequences of subspaces of $\mathcal{H}^2$. Our proofs employ function and
operator theoretic techniques in one and several variables; a variety of number
theoretic arguments are used throughout the paper in our study of special
classes of symbols $g$.
