  In this article, we introduce \textit{Mallows processes}, defined to be
continuous-time c\`adl\`ag processes with Mallows distributed marginals. We
show that such processes exist and that they can be restricted to have certain
natural properties. In particular, we prove that there exists \textit{regular}
Mallows processes, defined to have their inversions numbers
$\mathrm{Inv}_j(\sigma)=|\{i\in[j-1]:\sigma(i)>\sigma(j)\}|$ be independent
increasing stochastic processes with jumps of size $1$. We further show that
there exists a unique Markov process which is a regular Mallows process.
Finally, we study properties of regular Mallows processes and show various
results on the structure of these objects. Among others, we prove that the
graph structure related to regular Mallows processes looks like an
\textit{expanded hypercube} where we stacked $k$ hypercubes on the dimension
$k\in[n]$; we also prove that the first jumping times of regular Mallows
processes converge to a Poisson point process.
