  In this survey, we discuss a series of linearization problems--for Poisson
structures, Lie algebroids, and Lie groupoids. The last problem involves a
conjecture on the structure of proper groupoids. Attempting to prove this by
the method of averaging leads to problems concerning almost actions of compact
groups and almost invariant submanifolds for compact group actions. The paper
ends with a discussion of possible extensions of the convexity theorems for
momentum maps of hamiltonian actions of compact groups.
