The International GPS Service (IGS) now regularly makes accurate GPS ephemeris and satellite clock information available over the Internet. The satellite coordinates are given in the International Terrestrial Reference Frame (ITRF). This thesis investigates Precise Point Positioning (PPP) using dual and single-frequency pseudorange and carrier phase observations. Both the static and kinematic modes are investigated. The static PPP solution examples use six-hour data sets from four stations. The observations were made while Selective Availability (SA) was active and after it had been discontinued. The static solutions agree to within 10 cm with published coordinates or with solutions obtained from the Jet Propulsion Laboratory (M) PPP Internet service. The kinematic solutions show a discrepancy of less than one meter, mostly around half a meter. For observations with low multipath, the research shows that single-frequency ionosphere-free PPP solutions are equivalent to the dual-frequency solutions. In case of single-frequency observations the pseudorange dominates the solution. Using a priori tropospheric information does not seem to improve dual-frequency PPP solutions as compared to the case when the vertical tropospheric delay is estimated as part of the Kalman filter solution. However, a priori tropospheric information seems to provide benefits to single-frequency kinematic PPP. The Saastamoinen model is used when computing the zenith tropospheric delay. In all cases, the Neill's mapping function is applied. The studies show high correlation between receiver clock and the up coordinate. The troposphere has a high negative correlation with receiver clock and the up coordinate. However, the troposphere is more correlated with the receiver clock than the up component. All solutions incorporate corrections for solid earth tides, relativity, and satellite antenna phase center offsets. Corrections have not been applied for the phase wind-up angle. The widelane and ionospheric functions are used to detect and fix cycle slips in a semi-graphical manner. Since even a single cycle slip significantly falsifies PPP solutions, it is suggested that between-satellite carrier phases be used as another way of detecting slips (now since SA has been discontinued). The software consists mostly of highly modular Mathcad functions that form an excellent base for continued research of PPP.