A central task for theoretical physics is explaining how the strange choreography of the microscopic world - the quantum-mechanical dance of electrons - produces the multifarious characteristics of macroscopic stuff: metallicity, magnetism, superconductivity, and so on. Equally crucially, theory should reveal new macroscopic phenomena that have not yet been seen because we did not know to look for them. These are hard tasks, since an atom and a magnet (say) are separated by a staggering jump in scale and complexity. Fortunately, our understanding of quantum mechanics, as applied to assemblages of many interacting particles, is currently exploding. New types of quantum materials are emerging into the light which - unlike a simple magnet - have no analogue in classical physics. (They rely on the "spooky action at a distance" unique to quantum mechanics and ideas from topology - the mathematics of knots, etc.) Separate work has shown that the central assumptions of statistical mechanics fail radically in some strongly disordered (dirty) quantum systems. Simultaneously, we are discovering that phase transitions between different quantum states are far subtler than we thought. The unifying theme for this proposal is a very general picture of physical systems in terms of fluctuating extended objects - for example vortex lines, or flux lines, or 'worldlines' in space- time. Such geometric descriptions are often more useful than descriptions in terms of electrons. For example, certain exotic states (quantum 'spin liquids' and related 'topological paramagnets') are best viewed as as Schrodinger's-cat-like mixtures of different configurations of loops, representing flux lines in a field which emerges miraculously from the dance of the electrons. Using pictures like this, I will tackle such questions as: how do we describe the new types of quantum phase transition theoretically? What do they teach us about quantum field theory? How do we realise the theoretically predicted topological states? How robust are they to perturbations and disorder? These are crucial questions for theoretical physics, which we must answer in order to explain the diversity of material behaviours that emerge from the (deceptively simple) laws of quantum mechanics.