  In this dissertation, we estimate the population of different classes of BNS
systems that are visible to gravitational-wave observatories. Given that no
ultra-compact BNS systems have been discovered in pulsar radio surveys, we
place a 95\% confidence upper limit of $\sim$850 and $\sim$1100 ultra-compact
neutron star--white dwarf and double neutron star (DNS) systems that are
beaming towards the Earth, respectively. We show that among all of the current
radio pulsar surveys, the ones at the Arecibo radio telescope have the best
chance of detecting an ultra-compact BNS system. We also show that adopting a
survey integration time of $t_{\rm int} \sim 1$~min will maximize the
signal-to-noise ratio, and thus, the probability of detecting an ultra-compact
BNS system. Similarly, we use the sample of nine observed DNS systems to derive
a Galactic DNS merger rate of $\mathcal{R}_{\rm MW} =
37^{+24}_{-11}$~Myr$^{-1}$, where the errors represent 90\% confidence
intervals. Extrapolating this rate to the observable volume for LIGO, we derive
a merger detection rate of $\mathcal{R} = 1.9^{+1.2}_{-0.6} \times \left(D_{\rm
r}/100 \ \rm Mpc \right)^3 \rm yr^{-1}$, where $D_{\rm r}$ is the range
distance for LIGO. This rate is consistent with that derived using the DNS
mergers observed by LIGO. Finally, we measure the sense of rotation of the
older millisecond pulsar, pulsar A, in the DNS J0737--3039 system and find that
it rotates prograde with respect to its orbit. This is the first direct
measurement of the sense of rotation of a pulsar and a direct confirmation of
the rotating lighthouse model for pulsars. This result confirms that the spin
angular momentum vector is closely aligned with the orbital angular momentum,
suggesting that kick of the supernova producing the second born pulsar
J0737--3039B was small.
