The unstable nature of fluid motion is a classical problem whose
mathematical roots go back to the 19th Century. It has important applications
to many aspects of our life from such disparate issues as predicting the
weather to regulating blood flow. Instabilities might lead to turbulence or
to new nonlinear flows which themselves might become unstable. We will
discuss some of the mathematical techniques which can be used to gain
insight into fluid instabilities. These tools include nonlinear PDE,
spectral theory and dynamical systems.