Stem, root or leaf cuttings are the nursery standard for the propagation of perennials, especially those whose clumps grow woody with time. The benefit of this method is that the young plants are true clones of their parents.
Leaf cutting is the simplest and most miraculous of the methods. It works for African violets, hydrangeas and begonias and it consists of picking a leaf with a long stem and sticking it in the ground. That's all. If you happen to have rooting hormone, it doesn't hurt to dip the stem in it before planting. If not, honey will also work. If not, just add water. I didn't believe it either until I saw it for myself.
Stem cuttings are the most common, used for most of the woody perennials, including but not limited to roses, rosemary, lavender, fuchsias, mums and geraniums. Pick a sturdy stem that is woody but still green, cut it into four to six inch long pieces that contain at least one growth bud, bruise the end to kick start the plant's rooting process and plant it. Both the medium and the cutting must be kept consistently moist. Some people like to mist their plants, others prefer to cover them with a clear jar to create a greenhouse effect. Some plants take a long time to root, but you'll know if the process worked when the cutting starts sprouting new growth.
Root cuttings are used for the propagation of woody shrubs - lilacs, raspberries, Oregon grape hollies, mock oranges and Japanese anemones. Dig up roots that are at least pencil thin while the plant is dormant and cut sections three to six inches long with one or two growth nodes. After planting, water well to ensure the roots are well settled into the ground and there are no air pockets around them.
If you love root division, you'll be happy to know that it works for bulbs too, via scaling, slicing, scooping and scoring.
Scaling is a propagation method that seems almost custom designed for lilies, whose bulbs "bloom" naturally, turning them into tiny clusters that look like artichokes. Scaling lily bulbs is the easiest propagation method available, you just dig them up, tease the scales gently apart and replant them in the desired location.
I hesitate to dig up my lilies, because if they manage to make it through the winter without being eaten I consider it enough of a blessing and would not dream of jinxing my luck. It so happens that lily bulbs are a squirrel and rabbit food delicacy.
For onion shaped roots, like those of daffodils and hyacinths, slice the bulb vertically into equal pieces that have a small portion of the root attached, soak them in a fungicidal solution, place them in a closed zipper bag with moist vermiculite and keep them in a warm dark place for three months. When they start forming little bulbils at the end, they are ready to plant.
The most unusual method of bulb propagation is called scooping, or scoring, and it is used for hyacinths. Take a large dormant bulb, scoop out the basal plate without disturbing the scales, or alternately score and remove radial portions of it, and plant it in moist sand to about half the height of the bulb, with the scooped out portion facing up. Keep them in a warm, dark place and make sure the sand stays consistently moist. After three months the scooped out portion will be covered with bulbils, which can be easily detached and planted.