Nuclei of extremely high energies are persistently bombarding the Earth. The so-called ultra-high energy cosmic rays (UHECR) were discovered about 100 years ago. The world’s largest UHECR experiment for energies above 10^17 eV is the Pierre Auger Observatory in Argentina, sampling air showers during their passage through the atmosphere and at ground level. After more than ten years of operation it has accumulated a large amount of air shower measurements to tackle the open questions. Studies of charged primaries are complemented by searches for neutral primaries. Neutral primaries propagate on straight lines, pointing back to the location of their production. Although not observed yet, there might be a small fraction of primary photons of less than a few percent. UHE photons are expected to be created in resonant pion photoproductions of UHECR with photon backgrounds during their propagation. Alternatively, several exotic top-down models predict a significant photon fraction from decays of extremely heavy particles. Some top-down models predict a dominance of UHE photons above about 10^20 eV (100 EeV) and were already disfavoured by upper limits on the UHE photon flux and fraction. The main goal of this work is an update and improvement of the upper limits on UHE photons with the surface detector (SD) of the Pierre Auger Observatory. The SD becomes fully efficient for triggering photon-induced showers above 10 EeV. Air showers from primary hadrons and photons differ in their longitudinal development in the atmosphere as well as in their lateral distribution of the electromagnetic and muonic shower component at ground. As the SD does not measure these parameters directly in the standard data reconstruction, correlated SD observables are being investigated in order to establish a photon-hadron separation in the energy range above 10 EeV. Two parameters are combined in a multivariate analysis to optimize the separation power and to make use of the large SD aperture. Those parameters allow for a powerful compromise of high selection efficiency, which is found when optimizing the data selection towards large statistics preserving a sufficient reconstruction quality, and good photon-hadron seperation. An update of existing photon energy calibrations is presented and the separation parameters are studied carefully. Throughout the search of rare particles, a proper understanding of the electronics and possible faults or rare physical events is needed as they can create photon-like events. Electronics issues are discussed, such as direct light effects or PMT afterpulses. The main result of this work is a significant improvement of the last published limits, entering the region of photon-optimistic GZK-predictions. The upper limit (95% C.L.) on the diffusive, integral photon flux above 10 EeV for zenith angles between 30 degrees and 60 degrees is 1.9 x 10^-3 km^-2 sr^-1 yr^-1, corresponding to a maximum photon fraction of 0.72%. For 20 EeV (40 EeV) it is 0.99 x 10^-3 km^-2 sr^-1 yr^-1 (0.49 x 10^-3 km^-2 sr^-1 yr^-1 ), corresponding to a maximum photon fraction of 1.6% (6.17%). A differential upper limit has also been placed for the first time in the range of 10-30 EeV, with a maximum photon flux of 2.59 km^-2 sr^-1 yr^-1 and a maximum photon fraction of 2.73%. In this region it is most likely to observe a photon. Photon candidate events are being discussed.