Involutive Heegaard Floer homology is a variant on the 3-manifold invariant Heegaard Floer homology which incorporates the data of the conjugation symmetry on the Heegaard Floer complexes, and is in principle meant to correspond to $\mathbb{Z}_4$-Seiberg Witten Floer homology. It can be used to obtain two new invariants of homology cobordism and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We introduce involutive Heegaard Floer homology and its associated invariants and use it to give a new criterion for an element in the integer homology cobordism group to have infinite order, similar but not identical to a recent criterion given by Lin-Ruberman-Saviliev. Much of this talk is joint work with C. Manolescu; other parts are variously also joint with I. Zemke or with J. Hom and T. Lidman.