Results of 150 GHz continuum observations of several X-ray bright clusters of galaxies in the Sunyaev-Zel'dovich (SZ) effect, carried out using the APEX-SZ bolometer array on the APEX telescope, are presented. Follow-up observations of the cluster A2163 with the Large APEX Bolometer Camera (LABOCA) at 345 GHz on the same telescope have been carried out, yielding the largest map of a cluster of galaxies in the SZ increment to date. Details of possible contaminants affecting the SZ signal are outlined, and a simulated galaxy cluster survey is used to estimate levels of contamination from each source of emission. Radio point sources are found to be of possible concern, but more information on spectral slopes in the millimeter regime is needed before any decisive conclusions can be made. It is found that treating unresolved thermal sources as a source of excess noise yields an excellent approximation of this contaminant, which sets a fundamental confusion limit on single-frequency SZ measurements. The 1.4 GHz volume averaged radio source luminosity function is constructed from a large sample of galaxy clusters. In contrast to previous studies, an adaptive cluster volume is used to construct the luminosity function. This is shown to yield more robust results on volume averaged source counts, and for the first time a redshift evolution in this luminosity function is found, taking into account the variable effect on radio source confusion with redshift. Due to the large sample size, luminosity evolution and number density evolution can be separated. The APEX-SZ and LABOCA instruments are described from an observer's perspective. The analyses of pointing and calibration data are described, and the data quality of each instrument is assessed. The reduction of galaxy cluster data is described in detail. Losses in source brightness due to the removal of correlated noise are modeled with a transfer function based on the reduction of a point source convolved with the instrument beam. It is shown that this approach is valid for a wide range of source morphologies. Significant detections of 9 clusters of galaxies with APEX-SZ are presented. The clusters are modeled using the well-known isothermal beta-model. The SZ signals are found to be in overall agreement with predictions from X-ray measurements. The possibility of simultaneously extracting gas temperatures and bulk velocities from SZ measurements is briefly discussed. It is found that current data have poor leverage on the problem; reasonable constraints require accurate measurements both at high frequencies, where confusion due to dust emission becomes increasingly more of a problem, and at low frequencies, where radio point sources can contaminate the signal. A non-isothermal model of the galaxy cluster A2163 is presented, based on APEX-SZ and XMM-Newton X-ray data. Under the assumption of spherical symmetry, the two data sets are used to de-project the structure of the cluster in terms of temperature and density, and the results are used to derive the gas mass profile. The total mass profile is derived under the assumption of hydrostatic equilibrium, to allow for an estimate of the gas mass fraction. It is shown that the isothermal approximation well approximates the non-isothermal model inside the X-ray core radius. The mass profile from the non-isothermal analysis is in agreement with weak-lensing measurements. The APEX-SZ data is also used, in conjunction with LABOCA data at 345 GHz and previously published SZ data at other frequencies, to constrain the central Comptonization and bulk velocity of the cluster
