Many properties of a transformation $T$ of a measure space $(X,\mu)$ are expressed in terms of the spectral properties of the unitary oprator $U(\phi)=\phi\circ T$ on $L^2(X,\mu)$. The most typical and extreme examples are translations on a torus, with discrete spectrum, and Bernoulli shifts, with Lebesgue spectrum.
Let us call in a loose way quasiperiodic those dynamical features associated to irrational rotations, and hyperbolic those associated with Bernoulli shifts. The study of dynaimcal systems is mostly an effort to extend the scope of hyperbolic and quasiperiodic phenomena in order to cover "most" of the dynamics of "most" systems.
Starting with Siegel's linearization theorem in 1942, KAM theory deals with the persistence of quasiperiodic motions smoothly conjugated to irrational rotations. In the simplest case of holomorphic germs in one complex variable, Siegel's Diophantine condition was extended by Bruno; I proved that Bruno's condition is optimal. The geometric construction involved in the last result has been successfully generalized by Perez-Marco in order to get a grasp on some aspects of the non linearizable case.
In the related matter of analytic circle diffeomorphisms, the results are similar as long as the diffeomorphisms are small perturbations of rotations. But without this restriction an interesting phenomenon takes place: while in the $C^\infty$ category Herman's conjugacy theorem holds exactly under the expected Diophantine condition on the rotation number, the optimal arithmetic condition in the analytic case is more restrictive than Bruno's condition.
In higher dimensions, Arnold's linearization theorem of analytic perturbations of rotations holds under Bruno's condition, but the optimal condition is unknown, and probably different.
Recent results of Herman emphasize that the importance of Diophantine invaraint tori is far from restricted to the classical symplectic setting; based on a persistence theorem for codimension one invariant tori, he disproves the quasiergodic hypothesis and a conjecture of Pesin asserting the existence of a non zero Lyapunov exponent for volume preserving diffeomorphisms.
Many progresses have also been made in the study of "nonstandard" quasiperiodic motions. In the symplectic setting, Aubry-Mather invariant Cantori for twist diffeomorphisms have been generalized as minimal measures by Mather. Renormalization of complex quadratic polynomials is studied by Sullivan via a far-reaching generalization of Teichmüller's theory.
On the hyperbolic side, I will leave aisde the active field of interactions between Anosov flows and negative curvature, to concentrate on the merging concepts involving weak forms of hyperbolicity. The classical theory of uniformly hyperbolic compact invariant sets and uniformly hyperbolic diffeomorphisms was more or less complete around 1970; Oseledets theorem and Pesin's work provide us with the conceptual tools to study weaker forms of hyperbolicity.
The first significant breakthrough was due to Jakobson, who proved the existence of a positive Lyapunov exponent w.r.t. an absolutely continuous invariant measure for a positive measure set of parameters in the real quadratic family. The next breakthrough was achieved by Benedicks and Carleson for the Henon family, their results being extended by Mora and Viana for rather general surface diffeomorphisms. Along similar lines I proved that non infinitely renormalizable complex quadratic polynomials with all periodic orbits repulsive exhibit sufficient expansion to insure the local connectedness of their Julia sets; Lyubich and Shishikura went on to show that these Julia sets have measure 0.
To embed these examples in a more comprehensive setting, and to give a more precise and systematic account of their dynamics is a goal which should be reachable in the near future.
Finally, there is the fundamental question: how many dynamical systems are we able to understand through these methods? In the real quadratic family, hyperbolic polynoimals have been proved to be dense, by Swiatek and Lyubich. In the complex quadratic family, the same question boils down to the study of infinitely renormalizable polynomials, and the proof of the local connectedness of the Mandelbrodt set is probability not too far away. In higher dimensions, Palis conjecures that homoclinic bifurcations are dense amongst nonhyperbolic diffeomorphisms; this seems at the moment out of reach, but should be stimulating. The general approach which have been and should continue to be fruitful is the following: starting with a diffeomorphism for which the dynamics in some open set are well understood and exhibit a global bifurcation, one considers a finite dimensional family through this diffeomorphism and tries to understand the dynamics in this same open set for most values of the parameter (in the measure-theoretical sense). Fine geometric invariants, like the Hausdorff dimension of basic sets, are strongly related to this discussion.