I will introduce and discuss some recent development of a fundamental problem in topology - the classification of continuous maps between spheres up to homotopy. These mathematical objects are called homotopy groups of spheres. I will start with some geometric background - its connection to the Generalized Poincare Conjecture for example. I will then introduce some classical and new methods of doing such computations, using certain spectral sequences. If time permits, I will discuss some recent development using motivic homotopy theory, a theory that was designed to use algebraic topology to study algebraic geometry, but has now been applied successfully in the reverse direction. Old and new open problems will be mentioned along the discussion.