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The present thesis is a commencement of a generalization of covering results in specific settings, such as the Euclidean space or the sphere, to arbitrary compact metric spaces. In particular we consider coverings of compact metric spaces $(X,d)$ by balls of radius $r$. We are interested in the minimum number of such balls needed to cover $X$, denoted by $Ncal(X,r)$. For finite $X$ this problem coincides with an instance of the combinatorial textsc{set cover} problem, which is $mathrm{NP}$-complete. We illustrate approximation techniques based on the moment method of Lasserre for finite graphs and generalize these techniques to compact metric spaces $X$ to obtain upper and lower bounds for $Ncal(X,r)$. The upper bounds in this thesis follow from the application of a greedy algorithm on the space $X$. Its approximation quality is obtained by a generalization of the analysis of Chv'atal's algorithm for the weighted case of textsc{set cover}. We apply this greedy algorithm to the spherical case $X=S\^