We report here the non-detection of gravitational waves from the merger of binary neutron star systems and neutron-star--black-hole systems during the first observing run of Advanced LIGO. In particular we searched for gravitational wave signals from binary neutron star systems with component masses $in [1,3] M_{odot}$ and component dimensionless spins $< 0.05$. We also searched for neutron-star--black-hole systems with the same neutron star parameters, black hole mass $in [2,99] M_{odot}$ and no restriction on the black hole spin magnitude. We assess the sensitivity of the two LIGO detectors to these systems, and find that they could have detected the merger of binary neutron star systems with component mass distributions of $1.35pm0.13 M_{odot}$ at a volume-weighted average distance of $sim$ 70Mpc, and for neutron-star--black-hole systems with neutron star masses of $1.4M_odot$ and black hole masses of at least $5M_odot$, a volume-weighted average distance of at least $sim$ 110Mpc. From this we constrain with 90% confidence the merger rate to be less than 12,600 Gpc$^{-3}$yr$^{-1}$ for binary-neutron star systems and less than 3,600 Gpc$^{-3}$yr$^{-1}$ for neutron-star--black-hole systems. We find that if no detection of neutron-star binary mergers is made in the next two Advanced LIGO and Advanced Virgo observing runs we would place significant constraints on the merger rates. Finally, assuming a rate of $10^{+20}_{-7}$Gpc$^{-3}$yr$^{-1}$ short gamma ray bursts beamed towards the Earth and assuming that all short gamma-ray bursts have binary-neutron-star (neutron-star--black-hole) progenitors we can use our 90% confidence rate upper limits to constrain the beaming angle of the gamma-ray burst to be greater than ${2.3^{+1.7}_{-1.1}}^{circ}$ (${4.3^{+3.1}_{-1.9}}^{circ}$).
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