One of the best methods for plant propagation is called soil layering. All of the vining plants put out new roots at the leaf nodes. Sometimes these are just stumps, but often they can grow pretty long. Leading these to soil makes them grow into new root systems. It does not shock the plant the way that taking cuttings does because the old root system remains intact. It actually mimics the growing conditions of these plants in their native habit because they crawl along the ground putting down additional sets of roots at every leaf node.
A plant which does really well with this method is the Monstera Deliciosa; also called split leaf philodendron. It is a plant which can grow really large and leggy and refreshing the root system periodically is really good for the plant and helps it stay really healthy and vital. Plus, it is a really great way to create new plants to give away to your friends. (Or anyone who will take them off your hands. If you have good plant mojo you will soon be drowning in plants.)
Here is a monstera which has been layered and cut back several times. The pot is very crowded with roots, but that is ok as long as it is fertilized regularly.
These plants get so big that you cannot repot them with all their foliage, so the best way to maintain them is to soil layer all the old foliage for about a year or so, then cut the new growth away from the old pot and repot just the root ball.
The first step is to get a new pot big enough to contain all the old growth, and coil the old vines around in the new pot. The stems are fairly flexible, but they can be broken if they are bent too sharply. You need a pot big enough to hold all the stems you want to root, and big enough to contain them with gradual bends.
Once you have all the shoots from the old plant coiled into the new pot, you can also add any cuttings from another plant to help fill in the new pot. The cuttings can be long enough to simply stick vertically between the coiled stems. Monsteras root very quickly and easily. The shoots of the cuttings will lag behind the ones which were soil layered for a year or so, but will fill out the pot and make the plant very thick and bushy.
Once you have all the cuttings and stems in the new pot, all that is left to be done is fill it with potting mix. Most potting mixes these days are largely peat and will filter down among the stems. It takes a while to fill up the pot and get the mix settled in among the stems.
Then it needs to be watered thoroughly. When it is dry peat does not absorb water well, so several small waterings separated by 30-60 minutes to let the water penetrate works best.
This is what it looked like 5 months later. You can see the old pot and root mass behind it to the right in the corner. Since the old pot has very little foliage it does not need much light. Its main purpose is to provide nutrients to the plant through the old root system.
The best time to cut back a plant is early in the growth season - mid-March through May. This is the time when the plant has the easiest time putting out new sprouts. You can cut back a plant very severely at this time, and it will shock it into putting out a lot of new sprouts. Doing it later in the year, toward the end of summer or during autumn or early winter will stunt the plant and set it back. It may go several months without putting out new shoots, and the ones it does will not be as vital as the ones it puts on during the spring growing season.
Here is the other monstera that I cut back to fill in the pot shortly after I cut it back, and then a few months later.