We construct families of Lagrangian 3-torus fibrations resembling the topology of some of the singularities in \textit{Topological Mirror Symmetry} \cite{TMS}. We perform a detailed analysis of the affine structure on the base of these fibrations near their discriminant loci. This permits us to classify the aforementioned families up to fibre preserving symplectomorphism. The kind of degenerations we investigate give rise to a large number of symplectic invariants.

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin in \cite{Gua}, \cite{H3} (we call these ``generalized Lagrangian submanifolds'' in our paper), we introduce and study three other classes of submanifolds and their relationships. For generalized complex manifolds that arise from complex (resp., symplectic) manifolds, all three classes specialize to complex (resp., symplectic) submanifolds. In general, however, all three classes are distinct. We discuss some interesting features of our theory of submanifolds, and illustrate them with a few nontrivial examples. Along the way, we obtain a complete and explicit classification of all linear generalized complex structures. We then support our ``symplectic/Lagrangian viewpoint'' on the submanifolds introduced in \cite{Gua}, \cite{H3} by defining the ``generalized complex category'', modelled on the constructions of Guillemin-Sternberg \cite{GS} and Weinstein \cite{Wei2}. We argue that our approach may be useful for the quantization of generalized complex manifolds.

Floer cohomology is computed for the elements of the mapping class group of a surface $\Sigma$ of genus $g>1$ which are compositions of positive and negative Dehn twists along loops in $\Sigma$ forming a tree-pattern. The computations cover a certain class of pseudo-Anosov maps.

We prove that the vanishing spheres of the Lefschetz pencils constructed by Donaldson on symplectic manifolds of any dimension are conjugated under the action of the symplectomorphism group of the fiber. More precisely, a number of generalized Dehn twists may be used to conjugate those spheres. This implies the non-existence of homologically trivial vanishing spheres in these pencils. To develop the proof, we discuss some basic topological properties of the space of asymptotically holomorphic transverse sections.

Differential graded algebra invariants are constructed for Legendrian links in the $1$-jet space of the circle. In parallel to the theory for $\R^3$, Poincar\'e--Chekanov polynomials and characteristic algebras can be associated to such links. The theory is applied to distinguish various knots, as well as links that are closures of Legendrian versions of rational tangles. For a large number of two-component links, the Poincar\'e--Chekanov polynomials agree with the polynomials defined through the theory of generating functions. Examples are given of knots and links which differ by an even number of horizontal flypes that have the same polynomials but distinct characteristic algebras. Results obtainable from a Legendrian satellite construction are compared to results obtainable from the DGA and generating function techniques.