  Lagrangian cobordisms between Legendrian knots arise in Symplectic Field
Theory and impose an interesting and not well-understood relation on Legendrian
knots. There are some known "elementary" building blocks for Lagrangian
cobordisms that are smoothly the attachment of $0$- and $1$-handles. An
important question is whether every pair of non-empty Legendrians that are
related by a connected Lagrangian cobordism can be related by a ribbon
Lagrangian cobordism, in particular one that is "decomposable" into a
composition of these elementary building blocks. We will describe these and
other combinatorial building blocks as well as some geometric methods,
involving the theory of satellites, to construct Lagrangian cobordisms. We will
then survey some known results, derived through Heegaard Floer Homology and
contact surgery, that may provide a pathway to proving the existence of
nondecomposable (nonribbon) Lagrangian cobordisms.
