From P.A. Smith to the present. Around 80 years ago, Smith proved the remarkable result that if a finite $p$-group $G$ acts on a compact space $X$ that has the mod $p$ homology of a sphere, then the fixed point space $X^G$ also has the mod $p$ homology of a sphere. Equivariant algebraic topology has developed in fits and starts ever since. I'll give some glimpses of current directions and questions.
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