This thesis consists of a summary of two papers dealing with questions related to Legendrian submanifolds of contact manifolds together with exact Lagrangian cobordisms between Legendrian submanifolds. The focus is on studying Legendrian submanifolds from the perspective of their handle decompositions. The techniques used are mainly from Symplectic Field Theory. In Paper I, a series of examples of Legendrian surfaces in standard contact 5-space are studied. For every g > 0, we produce g+1 Legendrian surfaces of genus g, all with g+1 transverse Reeb chords, which lie in distinct Legendrian isotopy classes. For each g, exactly one of the constructed surfaces has a Legendrian contact homology algebra admitting an augmentation. Moreover, it is shown that the same surface is the only one admitting a generating family. Legendrian contact homology with Novikov coefficients is used to classify the different Legendrian surfaces. In particular, we study their augmentation varieties. In Paper II, the effect of a Legendrian ambient surgery on a Legendrian submanifold is studied. Given a Legendrian submanifold together which certain extra data, a Legendrian ambient surgery produces a Legendrian embedding of the manifold obtained by surgery on the original submanifold. The construction also provides an exact Lagrangian handle-attachment cobordism between the two submanifolds. The Legendrian contact homology of the submanifold produced by the Legendrian ambient surgery is then computed in terms of pseudo-holomorphic disks determined by data on the original submanifold. Also, the cobordism map induced by the exact Lagrangian handle attachment is computed. As a consequence, it is shown that a sub-critical standard Lagrangian handle attachment cobordism induces a one-to-one correspondence between the augmentations of the Legendrian contact homology algebras of its two ends.

This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on Legendrian contact homology. In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So to equip the trees with orientations corresponds to orienting the determinant line bundle of the dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We define an  orientation of this line bundle and prove that it is well-defined in the limit. We also prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing the orientation of the trees, and we give an explicit description of this algorithm. In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to prove invariance of Legendrian contact homology for Legendrian knots over the integers can be derived analytically.  This is proved using the orientation scheme from Paper I together with a count of abstractly perturbed flow trees  of the Lagrangian cobordisms. In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in the standard symplectic plane.

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