  The dynamics of open quantum systems (i.e., of quantum systems interacting
with an uncontrolled environment) forms the basis of numerous active areas of
research from quantum thermodynamics to quantum computing. One approach to
modeling open quantum systems is via a Collision Model. For instance, one could
model the environment as being composed of many small quantum systems
(ancillas) which interact with the target system sequentially, in a series of
"collisions". In this thesis I will discuss a novel method for constructing a
continuous-time master equation from the discrete-time dynamics given by any
such collision model. This new approach works for any interaction duration,
$\delta t$, by interpolating the dynamics between the time-points $t =
n\,\delta t$. I will contrast this with previous methods which only work in the
continuum limit (as $\delta t\to 0$). Moreover, I will show that any
continuum-limit-based approach will always yield unitary dynamics unless it is
fine-tuned in some way. For instance, it is common to find non-unitary dynamics
in the continuum limit by taking an (I will argue unphysical) divergence in the
interaction strengths, $g$, such that $g^2 \delta t$ is constant as $\delta t
\to 0$.
