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For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to $B$ along the zero section of the sheaf of relative differentials. The most interesting examples are when $B$ is $M_g$ or $A_g$, and the fibrations are the Jacobian fibration or the universal family of abelian varieties.

Is there a good reference for the properties of this bundle? And for the determinant line bundle on these spaces? Things like self-intersection numbers, cohomology (in particular global sections) are particularly of interest.

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    $\begingroup$ Although this question was asked a long time ago, I would like to advertise my recent paper with V. Maillot on the determinant line bundle (which answers some of the questions asked in the references given in Tony Pantev's answer) : "On the determinant bundles of abelian schemes". Compositio Math. 144 (2008), 495--502. $\endgroup$
    – Damian Rössler
    Commented Oct 2, 2011 at 10:04

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The list of references is way too long. Here are some classical texts containing both the setup and calculations:

1) P. Deligne, Le déterminant de la cohomologie, Current Trends in Arithmetical Algebraic Geometry, Contemp. Math., no. 67, AMS, Providence, 1987.

2) Gerd Faltings, Ching-Li Chai, Degenerations of abelian varieties, Springer-Verlag, 1990.

3) L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985).

And here are a couple of more recent papers that deal with the self-intersection and cohomology calculations:

4) Alexis Kouvidakis, Theta line bundles and the determinant of the Hodge bundle, arXiv:alg-geom/9604017.

5) Alexander Polishchuk, Determinant bundles for abelian schemes, arXiv:alg-geom/9703021.

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