  We study non-compact scaling limits of uniform random planar quadrangulations
with a boundary when their size tends to infinity. Depending on the asymptotic
behavior of the boundary size and the choice of the scaling factor, we observe
different limiting metric spaces. Among well-known objects like the Brownian
plane or the infinite continuum random tree, we construct two new one-parameter
families of metric spaces that appear as scaling limits: the Brownian
half-plane with skewness parameter $\theta$ and the infinite-volume Brownian
disk of perimeter $\sigma$. We also obtain various coupling and limit results
clarifying the relation between these objects.
