  In the Hospital Residents problem with lower and upper quotas ($HR-Q^U_L$),
the goal is to find a stable matching of residents to hospitals where the
number of residents matched to a hospital is either between its lower and upper
quota or zero [Bir\'o et al., TCS 2010]. We analyze this problem from a
parameterized perspective using several natural parameters such as the number
of hospitals and the number of residents. Moreover, we present a
polynomial-time algorithm that finds a stable matching if it exists on
instances with maximum lower quota two. Alongside $HR-Q^U_L$, we also consider
two closely related models of independent interest, namely, the special case of
$HR-Q^U_L$ where each hospital has only a lower quota but no upper quota and
the variation of $HR-Q^U_L$ where hospitals do not have preferences over
residents, which is also known as the House Allocation problem with lower and
upper quotas. Lastly, we investigate how the parameterized complexity of these
three models changes if preferences may contain ties.
