The moduli space of a certain class of geometric objects parameterizes the isomorphism classes of such objects.

Moduli spaces occur naturally in classification problems.

They appear in many branches of mathematics and in particular in algebraic geometry.

For a natural number $g>1$ the moduli space $M_g$ classifies smooth projective curves of genus $g$.

In 1969 Deligne and Mumford introduced a compactification of this space by means of stable curves.

The geometry of these moduli spaces have been studied extensively since then by people from different perspectives.

Many questions about the geometry of moduli spaces of curves involve the so called tautological classes.

These are the most natural algebraic cycles on these spaces.

In this talk I will review well-known facts and conjectures about tautological classes.

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