Topological states of matter - now what?

The interest in topological condensed matter dates back to the early days of the quantum Hall effect in the late 80's.  Further excitement has emerged since the discovery of time-reversal invariant topological systems in 2005 and has led to the Nobel prize of 2016, awarded to some of the field's pioneers - Kosterlitz, Thouless and Haldane.  The introduction of topology to the classification of states of matter is arguably one of the most important paradigm shifts of our time. We now understand that states of matter should be classified not only by their dimensionality and symmetries but also by their topological properties.  In this talk I will present a biased view of the advances in the field and the challenges ahead.  I will organize the discussion in three parts: fundamental, complex and applications.  The 'fundamental' part will include research done in the direction of organizing the possible phases of materials and some known examples.  The 'complex' part will focus on research into topological condensed matter systems beyond the clean, equilibrium and non-interacting limits and the 'applications' part will discuss driving, manipulating and realizing topological systems and possible applications.