The Arctic sea ice cover has a great influence on the climate and is believed to respond rapidly to climate changes. Since 2003 the Ice, Cloud and land Elevation Satellite (ICESat) laser altimetry mission has provided satellite altimetry over the ice covered Arctic Ocean up to 86 N. In this thesis, the main topic is to estimate the sea surface height in the Arctic Ocean from ICESat laser altimetry data and to use this information to estimate sea ice freeboard heights, gravity anomalies and mean dynamic topography. The laser altimeter measures the height of the surface topography, which in the Arctic is a combination of sea ice and open water. The sea surface height is found by a "lowest-level" filtering procedure, originally developed for airborne lidar measurements, which assumes that the lowest levels measured represent the open water in the ice pack. The sea surface obtained this way is used to estimate the sea ice freeboard, and shows good qualitative correlation to QuikSCAT scatterometer data. As the method depends on the presence of open water, the method is underestimating the sea ice freeboard heights, when compared to coincident high resolution airborne lidar measurements in areas with thick ice or ice of high concentration. Overall, a decrease in the mean freeboard heights of approximately 10 - 15 cm (corresponding to 70 - 75 cm in thickness) are observed, since the beginning of the ICESat observations in 2003. The potential for ICESat derived geoid and gravity anomalies are investigated. The ICESat gravity grid shows all the major tectonic features of the Arctic Ocean at high resolution. The results show that the laser altimetry data provides excellent gravity results comparable to open ocean altimetry even over the most heavy ice conditions. Subtracting a geoid model from the mean sea surface can be used to improve the knowledge of the sea surface topography (semi permanent circulation patterns). The dynamic topography observed by ICESat maps the main circulation in the Arctic, e.g. the Beaufort Gyre and the lower heights in the Norwegian-Greenland Sea. A comparison to existing global and regional oceanographic models of the Arctic mean dynamic topography shows good qualitative agreement.