Abstract: This talk will be accessible to graduate students of all levels. In my talk, I give an introduction to zeta functions associated to manifolds that are generalizations of the Riemann zeta function. I will give a general overview of how they are defined and what they are used for (for example, how they can used to understand geometrical aspects of manifolds). Then I will discuss new exotic phenomena of zeta-functions associated to conic manifolds (like cones in R^3). It turns out that the meromorphic extensions of these zeta-functions have very strange properties. In particular, they have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicities. I present simple examples illustrating these strange properties.