The scattering of electroweak gauge bosons is closely connected to the electroweak gauge symmetry and its spontaneous breaking through the Brout-Englert-Higgs mechanism. Since it contains triple and quartic gauge boson vertices, the measurement of this scattering process allows to probe the self-interactions of weak bosons. The contribution of the Higgs boson to the weak boson scattering amplitude ensures unitarity of the scattering matrix. Therefore, the scattering of massive electroweak gauge bosons is sensitive to deviations from the Standard Model prescription of the electroweak interaction and of the properties of the Higgs boson. At the Large Hadron Collider (LHC), the scattering of massive electroweak gauge bosons is accessible through the measurement of purely electroweak production of two jets and two gauge bosons. No such process has been observed before. Being the channel with the least amount of background from QCD-mediated production of the same final state, the most promising channel for the first measurement of a process containing massive electroweak gauge boson scattering is the one with two like-charge W bosons and two jets in the final state. This thesis presents the first measurement of electroweak production of two jets and two identically charged W bosons, which yields the first observation of a process with contributions from quartic gauge interactions of massive electroweak gauge bosons. An overview of the most important issues in Monte Carlo simulation of vector boson scattering processes with current Monte Carlo generators is given in this work. The measurement of the final state of two jets and two leptonically decaying same-charge W bosons is conducted based on proton-proton collision data with a center-of-mass energy of √s = 8 TeV, taken in 2012 with the ATLAS experiment at the LHC. The cross section of electroweak production of two jets and two like-charge W bosons is measured with a significance of 3.6 standard deviations to be σ(W± W±jj−EW[fiducial]) = 1.3 ± 0.4(stat.) ± 0.2(syst.) fb in a fiducial phase space region selected to enhance the contribution from W W scattering. The measurement is compatible with the Standard Model prediction of σ(W±W± jj−EW[fiducial]) = 0.95 ± 0.06 fb. Based on this measurement, limits on anomalous quartic gauge couplings are derived. The effect of anomalous quartic gauge couplings is simulated within the framework of an effective chiral Lagrangian unitarized with the K-matrix method. The limits for the anomalous coupling parameters α4 and α5 are found to be −0.14 < α4 < 0.16 and −0.23 < α5 < 0.24 at 95 % confidence level. Furthermore, the prospects for the measurement of the electroweak production of two same-charge W bosons and two jets within the Standard Model and with additional doubly charged resonances after the upgrade of the ATLAS detector and the LHC are investigated. For a high-luminosity LHC with a center-of-mass energy of √s = 14 TeV, the significance of the measurement with an integrated luminosity of 3000 fb^−1 is estimated to be 18.7 standard deviations. It can be improved by 30 % by extending the inner tracking detector of the atlas experiment up to an absolute pseudorapidity of |η| = 4.0.