  I develop and apply a Bayesian method for quantitatively estimating
properties of the quark-gluon plasma (QGP), an extremely hot and dense state of
fluid-like matter created in relativistic heavy-ion collisions.
  The QGP cannot be directly observed -- it is extraordinarily tiny and
ephemeral, about $10^{-14}$ meters in size and living $10^{-23}$ seconds before
freezing into discrete particles -- but it can be indirectly characterized by
matching the output of a computational collision model to experimental
observations. The model, which takes the QGP properties of interest as input
parameters, is calibrated to fit the experimental data, thereby extracting a
posterior probability distribution for the parameters.
  In this dissertation, I construct a specific computational model of heavy-ion
collisions and formulate the Bayesian parameter estimation method, which is
based on general statistical techniques. I then apply these tools to estimate
fundamental QGP properties, including its key transport coefficients and
characteristics of the initial state of heavy-ion collisions.
  Perhaps most notably, I report the most precise estimate to date of the
temperature-dependent specific shear viscosity $\eta/s$, the measurement of
which is a primary goal of heavy-ion physics. The estimated minimum value is
$\eta/s = 0.085_{-0.025}^{+0.026}$ (posterior median and 90% uncertainty),
remarkably close to the conjectured lower bound of $1/4\pi \simeq 0.08$. The
analysis also shows that $\eta/s$ likely increases slowly as a function of
temperature.
  Other estimated quantities include the temperature-dependent bulk viscosity
$\zeta/s$, the scaling of initial state entropy deposition, and the duration of
the pre-equilibrium stage that precedes QGP formation.
