This thesis is in the broad area of random conformal geometry, combining tools from probability and complex analysis.

We mainly consider Liouville quantum gravity (LQG), a model introduced in the physics literature in the 1980s by Polyakov in order to provide a canonical example of a random surface with conformal symmetries and formally given by the Riemannian metric tensor "$e\gamma h(dx2+dy2)$'' where $h$ is a Gaussian free field (GFF) on a planar domain and $\gamma \in (0,2)$. Duplantier and Sheffield constructed the $\gamma$-LQG area and boundary length measures, which fall under the framework of Kahane's Gaussian multiplicative chaos. Later, a conformally covariant distance metric associated to $\gamma$-LQG was constructed for whole-plane and zero-boundary GFFs.

In this thesis we describe the $\gamma$-LQG metric corresponding to a free-boundary GFF and derive basic properties and estimates for the boundary behaviour of the metric using GFF techniques. We use these to show that when one uses a conformal welding to glue together boundary segments of two appropriate independent LQG surfaces to get another LQG surface decorated by a Schramm--Loewner evolution (SLE) curve, the LQG metric on the resulting surface can be obtained as a natural metric space quotient of those on the two original surfaces. This generalizes results of Gwynne and Miller in the special case $\gamma =8/3$ (for which the LQG metric can be explicitly described in terms of Brownian motion) to the entire subcritical range $\gamma \in (0,2)$. Moreover, we show that LQG metrics are infinite-dimensional (in the sense of Assouad) and thus that their embeddings into the plane cannot be quasisymmetric.

We also consider chemical distance metrics associated to conformal loop ensembles, the loop version of SLE, using the imaginary geometry coupling to the GFF to bound the exponent governing the conformal symmetries of such a metric.