The topology of Lagrangian submanifolds is still mysterious today even after more than twenty five years of intensive investigation. Understanding this type of submanifold is motivated by many problems some originating in physics some others in real algebraic geometry as well as in other areas of mathematics. I will first review some of the standard results in the subject starting with Gromov and Floer and further pursued by Hofer, Salamon, Oh and many others. They all depend on detecting certain types of pseudoholomorphic curves with Lagrangian boundary conditions. Managing algebraically this type of curves is a hot topic of current research and I will describe some results in this direction obtained in joint work with Francois Lalonde as well as some obtained together with Paul Biran. I will also describe the way this is related to work of Fukaya-Oh-Ohta-Ono.

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