Abstract: Invariant harmonic analysis on a reductive algebraic group defined over a local field consists primarily in the study of two types of conjugation-invariant distributions: orbital integrals and characters of admissible representations. We will begin by introducing these objects and their relationships, along with some classical results at the foundation of the theory. For certain applications arising from the Langlands Program and comparison of trace formulas, it turns out that conjugation is too fine an equivalence relation to compare these objects across different groups. Stable conjugation is a coarser relation; two elements of the group are stably conjugate if (roughly speaking) they are conjugate over an algebraic closure of the base field. We will only sketch the main ideasbehind obtaining stable distributions from the characters of irreducible representations (Langlands packets and Arthur packets). This talk is intended for a general mathematical audience and will be accessible to non-specialists.