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The aim of this thesis is to study under which conditions $K3$ surfacesallowing a triple-point-free model satisfy the monodromy property. Thisproperty is a quantitative relation between the geometry of the degeneration ofa Calabi-Yau variety $X$ and the monodromy action on the cohomology of $X$: aCalabi-Yau variety $X$ satisfies the monodromy property if poles of the motiviczeta function $Z_{X,\omega}(T)$ induce monodromy eigenvalues on the cohomologyof $X$.  In this thesis, we focus on $K3$ surfaces allowing a triple-point-free model,i.e., $K3$ surfaces allowing a strict normal crossings model such that threeirreducible components of the special fiber never meet simultaneously. Crauderand Morrison classified these models into two main classes: so-called flowerpotdegenerations and chain degenerations. This classification is very precise,which allows to use a combination of geometrical and combinatorial techniquesto check the monodromy property in practice.  The first main result is an explicit computation of the poles of$Z_{X,\omega}(T)$ for a $K3$ surface $X$ allowing a triple-point-free model anda volume form $\omega$ on $X$. We show that the motivic zeta function can havemore than one pole. This is in contrast with previous results: so far, allCalabi-Yau varieties known to satisfy the monodromy property have a uniquepole. We prove that $K3$ surfaces allowing a flowerpot degeneration satisfy themonodromy property. We also show that the monodromy property holds for $K3$surfaces with a certain chain degeneration. We don't know whether all $K3$surfaces with a chain degeneration satisfy the monodromy property, and weinvestigate what characteristics a $K3$ surface $X$ not satisfying themonodromy property should have.

The monodromy property for K3 surfaces allowing a triple-point-free model