In this talk, I will survey on recent progresses on algebraic cycles, based in an essential way on Griffiths ideas in Hodge theory. The main object of study here is a Hodge structure, that is a complex vector space $H^k$ equipped with a decreasing filtration $F^pH^k$ subject to some rank conditions, and an integral structure $H^k\cong H^k_{\bf Z}\otimes {\bf C}$. Algebraic cycles can be studied using the "cycle map" $Z^p(X)\to F^p(H^{2p}(X)\cap H^{2p}(X,{\bf Z})$, and its refinement "Deligne-Abel-Jacobi cycle map" which takes into account a secondary class with values into the complex torus $J^{2p-1}(X)\coloneq H^{2p-1}(X)/F^pH^{2p-1}+H^{2p-1}(X,{\bf Z})$.
However, as shown by the lack of progresses on the Hodge conjecture, and on various conjectures on the kernel of the cycle map, these objects are not well understood, excepted for the codimension one case, where the exponential sequence is used to show the ideal correspondence between algebraic objects -- divisors modulo rational equivalence -- and Hodge theoretic objects -- the Hodge structure on $H^1$ and $H^2$.
Griffiths idea was to consider instead of the transcendental data -- the integral structure on cohomology -- another set of informations, still very rich, but purely algebraic, which is carried by the infinitesimal variation of Hodge structure. The new objects are now the bundle ${\scr H}^k$, which is equipped with the holomorphically varying Hodge filtration $F^p{\scr H}^k$; and the Gauss-Manin connection on it, which satisfies the transversality condition: $\nabla F^p\subset F^{p-1}\otimes\Omega$.
Algebraic cycles can be studied in this setting, and the analog of the transversality condition for the cycle map, called "horizontality", allows to construct infinitesimal invariants associated to families of algebraic cycles in families of varieties, from which one can deduce the following kind of statements:
The Abel-Jacobi map is trivial modulo torsion for the general member of certain families of varieties (e.g. hypersurfaces f degree $\geq 6$ in ${\bf P}^4$) (Green, Voisin).
A similar statement for the regulator map on surfaces (Bloch-Nori).
The non-triviality of the Abel-Jacobi map for general Calabi-Yau threefolds (Voisin).
The existence of cycles non algebraically equivalent to zero annihilated by the cycle map (Griffiths, Nori) and the non finite generation of the group of such cycles (Albano-Collino).
Criteria for ration equivalence of zero cycles on families of varieties, wtih application to the geometry of surfaces in ${\bf P}^3$ (Voisin).
Recovering a cycle from its infinitesimal invariant (Collino-Pirola).
Recently the subject has been beautifully revisited by Nori, whose work dominates a large part of the results listed above. Nori realized that the vanishing of certain infinitesimal invariants generalizes to a vanishing theorem for the relative cohomology $H(X\times S, Y_S)$ for $Y_S$ a complete family of sufficiently ample complete intersections parametrized by a smooth basis $S$. I will try to explain his "connectivity theorem", and the striking consequences it has on algeberaic cycles.