  The aim of this paper is to establish a lattice theoretical framework to
study the partially ordered set $\operatorname{\mathsf{tors}} A$ of torsion
classes over a finite-dimensional algebra $A$. We show that
$\operatorname{\mathsf{tors}} A$ is a complete lattice which enjoys very strong
properties, as bialgebraicity and complete semidistributivity. Thus its Hasse
quiver carries the important part of its structure, and we introduce the brick
labelling of its Hasse quiver and use it to study lattice congruences of
$\operatorname{\mathsf{tors}} A$. In particular, we give a
representation-theoretical interpretation of the so-called forcing order, and
we prove that $\operatorname{\mathsf{tors}} A$ is completely congruence
uniform. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}}
(A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is
called an algebraic quotient, and the corresponding lattice congruence is
called an algebraic congruence. The second part of this paper consists in
studying algebraic congruences. We characterize the arrows of the Hasse quiver
of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic
congruence in terms of the brick labelling. In the third part, we study in
detail the case of preprojective algebras $\Pi$, for which
$\operatorname{\mathsf{tors}} \Pi$ is the Weyl group endowed with the weak
order. In particular, we give a new, more representation theoretical proof of
the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian
lattice when $Q$ is a Dynkin quiver. We also prove that, in type $A$, the
algebraic quotients of $\operatorname{\mathsf{tors}} \Pi$ are exactly its
Hasse-regular lattice quotients.
