High energy proton-proton collision data collected at the Large Hadron Collider (LHC) have opened a possibility to study self-interactions, triple (TGC) and quartic (QGC) gauge couplings, of the electroweak vector-bosons of the Standard Model (SM). Electroweak processes that are dominated by the TGC and QGC have relatively low production cross sections at the LHC energy and are sensitive to possible new physics (NP) effects. Their precise measurements allow to test the previously unexplored area of vector-boson self-interactions of the SM and search for possible NP effects at the higher energy scale than it is accessible in direct searches of new particles and resonances at the LHC. Model-independent searches of the NP effects are focussed on measurements of anomalous triple and quartic gauge couplings, aTGC and aQGC, in the phase-space regions of the interesting electroweak processes that are optimised to the self-interaction amplitudes of the SM vector-bosons. The searches are usually performed within the frameworks of effective field theory (EFT) extensions of the SM. Electroweak processes sensitive to quartic gauge couplings via the vector-boson-scattering (VBS) are best suited to search for aQGC. Measurements of the VBS processes by the ATLAS detector at the LHC using proton-proton collision data collected at the centre-of-mass energy of 13 TeV are presented. Experimental methods as well as EFT frameworks of the measurements are reviewed. Future prospects of the VBS measurements at the LHC are also discussed.