Abstract In this chapter we will, as usual, begin by discussing some physical situations that are modelled by elliptic equations, as defined in a rather unfocused way in Chapter 3. Most of the examples involve scalar second-order equations, several of which are special cases, such as steady states, of evolution models discussed in Chapters 4 and 6. The methods we will use in the subsequent analysis of these models are more ad hoc than those used on hyperbolic equations, for the simple reason that we have no general well-posedness statement analogous to that for the Cauchy problem for hyperbolic equations. Moreover, we will find that the influence of the data for elliptic problems, especially singularities in the boundary data, is much less localised and ‘coherent’ than it is for hyperbolic equations, where we recall that many kinds of singularities merely propagate along characteristics.