From P.A. Smith theory to the Connor conjecture to the present. Around 1940, P. A. 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. Around 1960, Pierre Conner conjectured that if a compact Lie group $G$ acts on a space $X$, then under certain finiteness conditions the vanishing of the cohomology of $X$ implies the vanishing of the cohomology of the orbit space $X/G$. Equivariant algebraic topology has developed in fits and starts ever since. It has recently become one of the very most central areas of that subject. I'll give some glimpses of what equivariant cohomology is and how it applies to prove Smith theory and the Conner conjecture. I'll say just a little about current directions and questions.

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