Bay windows were originally found in the stately countryside manors of the English Renaissance. Most commonly these designs were installed at the end of a great hall, soaking either a ballroom or dining room in light while providing a view of the rolling lawns typical of these sprawling estates.
Because of their regal appeal, bay windows soon became implemented in all manner of residential architecture, from the countryside to the city, to add excess drama to any space.
Since the purpose of these windows is to make a big statement, great care must go into choosing an appropriate treatment that isn't overpowering.
Bay windows let in more light than most designs, so it makes sense to utilize this feature by not using heavy fabrics to block those rays. A sheer curtain will allow the light to shine through and blanket the room in a softer, more delicate glow.
There are other interesting ways to play with the unique design of a bay window. These windows are split into three sections, with the two bookending sections being called the returns. By choosing to alternate styles between the central portion of the window and the two returns, you give the window an interesting symmetry. Consider using sheer curtains on all three portions, but layer the returns with a heavier fabric, creating an effect that frames the central pane.
If your house has a bay window, it's probably because there is a view worth witnessing on a daily basis outside. Let that view and your bay window be the star of your design scheme. By choosing a simple, softer fabric without too showy a pattern or palette, the function of the bay window is better served. You don't want to hide such an envied architectural feature.