Definition. Given a property of mathematical structures of a given class, we say that $\kappa$ is a compact cardinal for the given property, if every structure of our class has the property, given that every ``smaller'' substructure has the property. $\kappa$ is a weakly compact cardinal for the property if the above holds for structures of ``size'' $\le \kappa$.

In the above, ``smaller'' and ``size'' may have different meanings for different properties. The (weak) compactness spectrum of the property is the class of cardinals that are (weakly) compact for the given property.

Since any compactness is related to large cardinals properties, the compactness spectra depends very much on the universe of set theory. So the question is really: ``which cardinals can consistently be compact for the given property''?

For many properties it will be interesting the analyse the possible compactness spectra for these properties. Typical examples are:

A group being free.

An abelian group being free.

A collection of countable sets having a transversal (a one-to-one choice function).

A collection of countable sets that can be disjointing. (Namely that one can remove a finite subset from each of the members of the family such that the resulting family will be mutually disjoint.)

A topological space being collectionwise Hausdorff.

A compact space being Corson. (Namely being homeomorphic to a subspace $X \subseteq [0,1]^\kappa$ such that for all $x \in X$ $\{\xi < \kappa | x(\xi) \ne 0\}$ is countable.)

In the talk we shall survey some of the known results about compactness spectra, as well as many open problems.