  The most commonly known triangulated categories arise from chain complexes in
an abelian category by passing to chain homotopy classes or inverting
quasi-isomorphisms. Such examples are called `algebraic' because they originate
from abelian (or at least additive) categories. Stable homotopy theory produces
examples of triangulated categories by quite different means, and in this
context the source categories are usually very `non-additive' before passing to
homotopy classes of morphisms. Because of their origin I refer to these
examples as `topological triangulated categories'.
  In these extended talk notes I explain some systematic differences between
these two kinds of triangulated categories. There are certain properties --
defined entirely in terms of the triangulated structure -- which hold in all
algebraic examples, but which fail in some topological ones. These differences
are all torsion phenomena, and rationally there is no difference between
algebraic and topological triangulated categories.

Algebraic Topology