  In this manuscript we present a pedagogical introduction to continuously
monitored quantum systems. We start by giving a simplified derivation of the
Markovian master equation in Lindblad form, in the spirit of collision models
and input-output theory, which describes the unconditional dynamics of a
continuously monitored system. The same formalism is then exploited to derive
stochastic master equations that describe the conditional dynamics. We focus on
the two most paradigmatic examples of continuous monitoring: continuous
photodetection, leading to a discontinuous dynamics with "quantum jumps", and
continuous homodyne measurements, leading to a diffusive dynamics. We then
present a derivation of feedback master equations that describe the dynamics
(either conditional or unconditional) when the continuous measurement
photocurrents are fed back to the system as a linear driving Hamiltonian, a
paradigm known as linear Markovian feedback. In the second part of the
manuscript we focus on continuous-variable Gaussian systems: we first present
the equations for first and second moments describing the dynamics under
continuous general-dyne measurements, and we then discuss in more detail the
conditional and unconditional dynamics under Markovian and state-based
feedback.
