In this overview talk we discuss several results regarding the dynamics of stochastic systems arising in or motivated by fluid mechanics. First, we discuss proving "Lagrangian chaos" in stochastic fluid mechanics, that is, demonstrating a positive Lyapunov exponent for the motion of a particle in the velocity field arising from the stochastic Navier-Stokes equations. We describe how this chaos can be used to deduce qualitatively optimal almost-sure exponential mixing of passive scalars, which can be used to provide a rigorous derivation of the power spectrum of passive scalar turbulence in certain regimes. Next, we describe more recently developed methods for obtaining strictly positive lower bounds and some quantitative estimates on the top Lyapunov exponent of weakly-damped stochastic differential equations, such as Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (called "Eulerian chaos" in fluid mechanics). We discuss upcoming work which combines many of the above ideas to study "symmetry breaking" in Lorenz-96 with degenerate forcing, giving an example of non-uniqueness of stationary measures (while providing a unique "physically correct" measure). Related ideas regarding nonlinear energy transfer in degenerately damped systems will also be discussed if time permits. All of the work except for the last (joint with Kyle Liss) is joint with Alex Blumenthal and Sam Punshon-Smith.