I will present two Cartesian grid based evaluation methods for boundary integrals. The first method uses intersection points of an underlying Cartesian grid with the boundary (curve or surface) as quadrature points. The corresponding numerical quadrature has super-algebraic convergence as long as the integrand function and the boundary curve/surface are smooth. The second method evaluates a boundary integral by solving an equivalent but simple interface problem on a Cartesian grid. It also works with intersection points of an underlying Cartesian grid with the boundary curve/surface but use them for discretization and correction only. The second approach is extendable for more general boundary integrals. Numerical experiments with the methods for examples in both two and three space dimensions will be presented.