How do I paint a tiled image, for example as a background of a panel?

The TexturePaint class can be used to achieve that.

public void paintComponent(Graphics g)
{
if(paint == null)
{
try
{
// Create TexturePaint instance the first time

int height = image.getHeight(this);
int width = image.getWidth(this);
bi = (BufferedImage) createImage(width, height);
Graphics2D biG2d = (Graphics2D) bi.getGraphics();
biG2d.drawImage(image, 0, 0, Color.black, this);
paint = new TexturePaint(bi,
new Rectangle(0,0,width,height));
}
catch (Exception ex)
{
ex.printStackTrace();
}
}

// Now actually do the painting