Topological order is a new kind of collective order beyond Landau's symmetry-breaking classification. Interesting in its own right, certain topologically ordered materials including the "chiral p-wave" superconductors and "non-Abelian" fractional quantum Hall (FQH) states may be used to realize fault-tolerant "topological" quantum computers. We have used both optical and electrical techniques to identify topological phases and to study their competition with broken symmetry orders. As the first example, I will describe ultra-sensitive magneto-optic Kerr measurements on ruthenate superconductor Sr2RuO4 with a recently developed "loop-less" fiber-optic Sagnac interferometer, identifying Sr2RuO4 to be a "chiral p-wave" topological superconductor. As a second example, I will discuss in 2D electrons at filling factor 5/2 the intriguing competition between the "non-Abelian" topological FQH state, an electronic liquid crystal phase and a newly discovered "reentrant isotropic compressible" state. I will also provide evidence for a novel rotational-symmetry-breaking FQH state as a consequence of this competition.