In the first measurement reported in this thesis, the 35Cl(n,gamma)36Cl reaction cross section in the resolved resonance region has been determined by measuring a NaCl sample (99.635% purity, 3.174x10^-3 35Cl atoms/barn) at the 185 m beam-line at the n_TOF time-of-flight facility at CERN using the total energy detection method, with a set-up composed of C6D6 detectors. The systematic uncertainty associated with the measured reaction yield was 4.4% up to 10 keV and 5.6% for larger energies up to around 100 keV, and resonances were analysed with the R-matrix code SAMMY up to an energy of around 60 keV. The measured resonance kernels are systematically larger than those reported in the ENDF/B-VIII.0 and JEFF-3.3 libraries by around 15%, and are moreover in agreement with one of the previous two measurements suggesting that a new evaluation should adopt this larger normalisation in the resonance region. The resonant component of the 30 keV Maxwellian averaged cross section (MACS) is also around 15% larger than that according to ENDF/BVIII.0, and is in reasonable agreement with a recent dedicated AMS measurement to within the 1-sigma level. For the second measurement reported in this thesis, ampoules of amorphous 99.5% enriched 13C were irradiated at the PF1b neutron beam line at the high-flux Institut LaueâLangevin (ILL) research reactor in order to produce 14C atoms. The precise ratio of 14C/13C was subsequently measured at the Vienna Environmental Research Accelerator (VERA) via accelerator mass spectrometry (AMS), allowing the 13C(n,gamma)14C thermal cross section to be accurately determined. This is the first measurement of this cross section at sub-eV energies via this technique and the result of 1.52 +/- 0.07 mb for the thermal cross section is in good agreement with other recent measurements which were performed via Prompt Gamma-ray Activation Analysis.