Digital images created by scanning small older photographs often suffer from conspicuous dust and scratches. The GIMP has a filter to correct this problem but as usual the filter should be used with moderation for it also causes the photograph to lose some fine detail. The dust in the following picture (the sun setting behind one of my students in Borneo) is very conspicuous against the smoothly graduated colours of the sunset. 

Choose Filters>Enhance>Despeckle and then play with the settings to achieve the best reduction of dust without affecting the student's lovely profile. Switch the preview on and off to see what effect your settings are having on the image.

To be honest, the Despeckle filter in the GIMP is much less effective than the 'dust and scratches' filter in Photoshop, but you can still get a worthwhile improvement with settings close to these, which I therefore recommend as the starting point for your own experimentation (note that Recursive is off):

Don't forget that you can apply the filter repeatedly if necessary.

You can get a spectacular reduction in dust and scratches with the following settings but at the expense of an excessive loss of detail and an artefact at the bottom of the picture (note that Recursive is on):

The radius determines how far the filter searches for differences among pixels, so use the smallest radius value that still has a good effect on reducing the dust defects. (Increasing the radius makes the image blurry.)

Don't worry if some defects remain after your restrained use of the Dust and Scratches filter - there is another way of tackling the worst defects: the clone tool (called the 'rubber stamp tool' in Photoshop). This is actually a very interesting tool, useful in many contexts in addition to this one. The clone tool copies parts of the image into neighboring parts using a brush.

First, select the clone tool in the main GIMP window by clicking it. Check that the tool's options (in the dialog below the toolbox) show that 'image source' and 'aligned' are selected. Then display the brush dialog with Dialogs>Brushes or Shift+Ctrl+B:

Choose the biggest built-in 'circle fuzzy' brush as shown here. Locate a defect somewhere in the picture then hold down the Ctrl key and click about 1cm left of the defect. Then release the Ctrl key and click or 'paint' across the defect. As you do so, you will be copying a part of the image 1 cm left of where you 'paint' since this is the relative position that you pre-defined by Ctrl-clicking. Of course, you do not have to copy from 1cm to the left. You can copy from above or below or anywhere - it all depends on the relative positions of the Ctrl-Click and the subsequent click or drag. This tool is a little tricky to master, but it's well worth persisting. When you've done your best at removing the defects from the above image, save it.

Before you continue, find out for yourself the difference between 'aligned' and 'non-aligned' in the options of the clone tool. Set the option back to 'aligned' before continuing.

Now that you have had some practice with the clone tool, try using it to turn this photo from Bali, Indonesia...

into this...

First, of course, you'll have to find the two differences between the two images...

I cheated a bit and used, in addition to the clone tool, the convolve (blur and sharpen) tool to sharpen the parts of the image that I had worked on. This might also be a good time to try the smudge tool , but I think you'll find the clone tool more satisfying to use. See the GIMP help pages for advice in using the blur and sharpen tool and the smudge tool.