The prime sources of gravitational radiation can be modeled as

isolated systems in which the waves propagate to infinity. However,

most numerical simulations of such sources introduce an artificial

outer boundary at a finite distance. There has been recent progress in

formulating boundary conditions for the Einstein equations which lead

to a constraint-preserving well-posed initial-boundary value problem.

I will review the basic ideas governing well-posedness of the

initial-boundary value problem for hyperbolic systems of second-order

wave equations, such as the harmonic formulation of Einstein's

equations and Maxwell's equations in the Lorentz gauge. For the

harmonic Einstein's equations, there are several options akin to

Sommerfeld boundary conditions. There remains the problem of

assigning appropriate boundary data. Unless the numerical evolution

is matched to an exterior solution, the only possibility is to give

homogeneous (vanishing) boundary data which approximate the asymptotic

behavior at infinity. One proposal is to use vanishing curvature data

(Psi_0 = 0 in the Newman-Penrose formalism). However, this results in

a second differential order boundary condition that does not fit the

theorems for well-posedness of the harmonic problem. I discuss how

Psi_0 = 0 data can be reformulated as first order data in Sommerfeld

form, the resulting deficiencies and alternative approaches.