  A subset of a topological space is said to be \emph{universally measurable}
if it is measured by the completion of each countably additive $\sigma$-finite
Borel measure on the space, and \emph{universally null} if it has measure zero
for each such atomless measure. In 1908, Hausdorff proved that there exist
universally null sets of real numbers of cardinality $\aleph_{1}$, and thus
that there exist at least $2^{\aleph_{1}}$ such sets. Laver showed in the
1970's that consistently there are just continuum many universally null sets of
reals. The question of whether there exist more than continuum many universally
measurable sets of reals was asked by Mauldin in 1978. We show that
consistently there exist only continuum many universally measurable sets. This
result also follows from work of Ciesielski and Pawlikowski on the iterated
Sacks model. In the models we consider (forcing extensions by suitably-sized
random algebras) every set of reals is universally measurable if and only if it
and its complement are unions of ground model continuum many Borel sets.
