  We present a unified approach to characterising fast-reaction limits of
systems of either two reaction-diffusion equations, or one reaction-diffusion
equation and one ordinary differential equation, on unbounded domains,
motivated by models of fast chemical reactions where either one or both
reactant(s) is/are mobile. For appropriate initial data, solutions of four
classes of problems each converge in the fast-reaction limit $k \to \infty$ to
a self-similar limit profile that has one of four forms, depending on how many
components diffuse and whether the spatial domain is a half or whole line. For
fixed $k$, long-time convergence to these same self-similar profiles is also
established, thanks to a scaling argument of Kamin. Our results generalise
earlier work of Hilhorst, van der Hout and Peletier to a much wider class of
problems, and provide a quantitative description of the penetration of one
substance into another in both the fast-reaction and long-time regimes.
