  Causal models provide rich descriptions of complex systems as sets of
mechanisms by which each variable is influenced by its direct causes. They
support reasoning about manipulating parts of the system and thus hold promise
for addressing some of the open challenges of artificial intelligence (AI),
such as planning, transferring knowledge in changing environments, or
robustness to distribution shifts. However, a key obstacle to more widespread
use of causal models in AI is the requirement that the relevant variables be
specified a priori, which is typically not the case for the high-dimensional,
unstructured data processed by modern AI systems. At the same time, machine
learning (ML) has proven quite successful at automatically extracting useful
and compact representations of such complex data. Causal representation
learning (CRL) aims to combine the core strengths of ML and causality by
learning representations in the form of latent variables endowed with causal
model semantics.
  In this thesis, we study and present new results for different CRL settings.
A central theme is the question of identifiability: Given infinite data, when
are representations satisfying the same learning objective guaranteed to be
equivalent? This is an important prerequisite for CRL, as it formally
characterises if and when a learning task is, at least in principle, feasible.
Since learning causal models, even without a representation learning component,
is notoriously difficult, we require additional assumptions on the model class
or rich data beyond the classical i.i.d. setting. By partially characterising
identifiability for different settings, this thesis investigates what is
possible for CRL without direct supervision, and thus contributes to its
theoretical foundations. Ideally, the developed insights can help inform data
collection practices or inspire the design of new practical estimation methods.
