The study of light nuclei is one of the most active areas in the nuclear physics today. Nuclei in the mass region A < 20 have been used as a testing sample for different theoretical models since the early days of nuclear physics. Nowadays the advent of ab initio calculations in various frameworks and the astrophysical interest makes the precise experimental data even more important. One of the topics of interest in the structure of light nuclei are the neutron-rich oxygen isotopes, starting already with astrophysically important 17O. One of the challenging questions in this field of research is how does the well developed cluster structure in 16O change with addition of neutrons in heavier oxygen isotopes. Therefore, the topic of this work is the study of the cluster-structure of the 17O and 18O. In this work results of two experiments are presented: the 13C+4He resonant elastic scattering on a thick gas target measurement and the 13C+9Be resonant particle spectroscopy measurement at E13C = 72 MeV. Both measurements were used due to their selectivity in populating the cluster-state candidates in 17O and 18O. The known 13C(4He,4He) elastic scattering cross-section has been extended both in angle (to the 175∘ in direct kinematics) and energy (up to Ex(17O)=13.8 MeV). The measured cross-section can be reproduced by using published set of parameters from the comprehensive Rmatrix fit. The new part of the spectrum contains several resonances of interest, for which the tentative values of spin, parity and Wigner ratio were determined using a simplified R-matrix fitting method. The resonant particle spectroscopy study of the 13C+4He→ 13C+ 4He+ 5He reaction has provided an improvement in statistics and the energy range covered by the previous study. Resonances decaying to the 13C*+4He decay channel were observed as well. The results of both measurements combined point to the pronounced a-cluster nature of the 8.9 MeV, 9.15 MeV and the 13.57 MeV 17O excited states. This is in accordance with predictions of the 17O rotational band based on the weakly coupled neutron to the well-known a-cluster structure in the 16O. However, precise determination of exact spins and parities, as well as partial and total widths are crucial to confirm the existence of the proposed rotational band. The results for the 18O nucleus were obtained from the 13C+9Be reaction measurements using the resonant particle spectroscopy method. The decays of the 18O* nucleus to the 14C+4He and 14C*+4He channels were studied, confirming the results and extending the excitation energy range of previous studies, as well as observing the a-decay of highly excited states for the first time. The results suggest that the 18O states decaying to the 14C+4He channel do not decay to the 14C*+4He channel and vice-versa, possibly due to a different structure. The 6He decay of the 18O has been observed for the first time, at an excitation energy Ex(18O) ≈ 26:5 MeV. Additional studies are necessary to assess the possibility of the 12C+2n+4He molecular structure of the 18O nucleus. In the study of the 13C+9Be→ 10Be+ 8Begs+ 4He reaction a few possible peaks that could indicate the 10Be+8Begs decay of the 18O nucleus were observed, but statistics was too low to obtain conclusive results. Therefore, additional measurements have to be performed. The resonant particle spectroscopy method may be used to collect the high quality data that would allow to compare the partial widths of the states, as well as to extract additional information using a DWBA calculation. The resonant elastic scattering may be used in order to clarify the spin and parity assignments of the cluster or molecular state candidates at higher 18O excitation energies, but the data on all the relevant inelastic channels have to be measured as well.