  We study the abstract problem of rounding fractional bipartite $b$-matchings
online. The input to the problem is an unknown fractional bipartite
$b$-matching, exposed node-by-node on one side. The objective is to maximize
the \emph{rounding ratio} of the output matching $\mathcal{M}$, which is the
minimum over all fractional $b$-matchings $\mathbf{x}$, and edges $e$, of the
ratio $\Pr[e\in \mathcal{M}]/x_e$. In offline settings, many dependent rounding
schemes achieving a ratio of one and strong negative correlation properties are
known (e.g., Gandhi et al., J.ACM'06 and Chekuri et al., FOCS'10), and have
found numerous applications. Motivated by online applications, we present
\emph{online dependent-rounding schemes} (ODRSes) for $b$-matching.
  For the special case of uniform matroids (single offline node), we present a
simple online algorithm with a rounding ratio of one. Interestingly, we show
that our algorithm yields \emph{the same distribution} as its classic offline
counterpart, pivotal sampling (Srinivasan, FOCS'01), and so inherits the
latter's strong correlation properties. In arbitrary bipartite graphs, an
online rounding ratio of one is impossible, and we show that a combination of
our uniform matroid ODRS with repeated invocations of \emph{offline} contention
resolution schemes (CRSes) yields a rounding ratio of $1-1/e\approx 0.632$. Our
main technical contribution is an ODRS breaking this pervasive bound, yielding
rounding ratios of $0.646$ and $0.652$ for $b$-matchings and simple matchings,
respectively. We obtain these results by grouping nodes and using CRSes for
negatively-correlated distributions, together with a new method we call
\emph{group discount and individual markup}, analyzed using the theory of
negative association. We present a number of applications of our ODRSes to
online edge coloring, several stochastic optimization problems, and algorithmic
fairness.
