EDIT: Tyler Lawson's answer was so nice that I was inspired to rewrite the notes discussed below to use Bredon homology in the definition of the Smith special homology groups. The original version is available for readers who prefer being low-tech; just click on the abstract button and a download button for this will become available.

Let $p$ be a prime, let $G$ be a finite $p$-group, and let $X$ be a "reasonable" finite-dimensional $G$-space (let me be vague about what reasonable means, but certainly a finite-dimensional simplicial complex upon which $G$ acts simplicially counts). Around 1940, P. Smith proved a number of theorems that relate the algebraic topology of $X$ to the algebraic topology of the fixed point set $X^G$. For instance, one of his theorems says that if $X$ is mod-$p$ acyclic, then so is $X^G$ (which implies in particular that $X^G$ is nonempty). I have notes on Smith theory on my page of notes here that give more details.

The original proofs of the main results of Smith theory use the "special homology groups" of Smith, which are defined algebraically and whose geometric meaning seems to me to be quite mysterious. I describe these homology groups in the notes above. When Borel introduced equivariant homology via the Borel construction, one of his original applications was to give a new proof of Smith's main results. These days, equivariant homology (both in the original Borel flavor and also using Bredon's more powerful definitions) is one of the fundamental tools in the theory of transformation groups.

**Question**: Is there a way to relate Smith's special homology groups to these modern developments? How are they related to equivariant homology? From my reading of the literature, they seem to have been mostly discarded from the toolbox of people working in the subject. Is it now understood that whatever information they contain is also contained in other places?