Brian Bogg's article from Fine Woodworking gives a good overview of the different methods for making curves and the pros and cons of each.
Perhaps the most straightforward option is to saw the curved shape out of solid wood. It is not the most material-efficient method, and may have areas of short grain that are weak, depending on the shape and orientation.
The Rex chair from Claude Home has sawn curves in the arms and back posts
cutting a curved part on the bandsaw
Cutting a curved and tapered part on the bandsaw
Curved forms can be built up from an underlying frame or armature that is skinned with slats or thin sheets. This works best for simple, not compound, curves. There are specialty plywoods known as "bendy ply" or "wiggle board" that can conform to very tight curves.
Sources:
Playful Furniture by Judson Beaumont in Fine Woodworking #217–Jan/Feb 2011 Issue
Judson Beaumont's build process: thin, flexible plywood secured to MDF frame
Cindy Cabinet by Judson Beaumont
Captain Ronnie's Desk by Katie Hudnall, 2007. Curves are made up of tonge-and-groove slats screwed to plywood frame.
Kerfing involves cutting multiple closely-spaced slices that allow an otherwise rigid piece of wood to bend. Plywood and solid wood can both be kerf-bent. Determining the spacing and the depth of the cuts to make the desired curve usually takes some experimenting. This online calculator is a great place to start. Kerf bends can be left exposed and visible, or covered with veneer to look like a solid panel.
Sources:
David Haig's take on curved panels.
Philip Morley's wall cabinet includes kerf bent sides.
David Haig creates bent panels by kerfing a solid piece of wood and veneering it.
David Haig's curved side table
Kerfed lining in a guitar, Jaén Guitars
Bent laminations take advantage of wood's natural ability to bend. By slicing thin, flexible plies from a larger board, each separate piece can conform to the desired curve. Gluing them back together against a form creates a strong, solid piece that has continuous grain throughout.
Sources:
Chris Gochnour's Guide to Bent Laminations on Fine Woodworking
The Wood Whisperer's video about bent laminations
Sheets of veneer laid over bending form
2. Vacuum press provides the clamping force while gluing the veneer sheets together
3. Finished panel retains the curved shape once the glue dries
Coopering is a barrel-making technique that builds up curved forms from staves with angled sides. It can be used to make open or closed forms, and simple or compound curves.
Sources:
Coopered Containers by Peter Lutz in Fine Woodworking #271-Nov/Dec 2018 Issue
Coopering a Door by Garrett Hack in Fine Woodworking #126–Sept/Oct 1997 Issue
Damion Fauser's walkthrough of making a coopered, tapered vase
Woodcraft's guide to building a coopered leg table
Coopered leg table exploded view
cradle for gluing coopered staves
Peter Lutz's trays are coopered forms made up of staves with angled sides.
Peter Lutz coopered trays
Cleer by Chelsea Witt. Ash with coopered legs.
Coopered bucket by master cooper Marshall Scheetz from Mortise & Tenon Magazine
Curio by Yuri Kobayashi
Coopered chest
Steam bending is the process of softening wood with steam until it becomes pliable enough to bend. The infusion of steam softens the wood’s lignins, the polymers that hold the cellulose fibers together. When the wood cools, the lignin hardens again, and the wood holds the bent shape. In order to steam bend, you will need:
species of wood amenable to bending
steam box
bending form--usually with lots of clamps, and maybe even a bending strap (helps prevent blowout on especially tight curves)
time, patience and spare materials--be prepared for failures and troubleshooting along the way!
Not all wood species are equally bendable. In general, temperate zone (domestic) hardwoods are the best, including oak (red or white), elm, hickory, ash, beech, birch, cherry, maple, and walnut. There’s dispute among woodworkers about whether softwoods (pine, cedar, etc.) are bendable, so it’s probably best to stay away from them. Similarly, most tropical hardwoods except mahogany are too dense to bend.
The wood should have as straight a grain as possible because it doesn’t bed easily around knots or circular grain. And if you have a choice between green wood and kiln dried, choose green. Kiln dried wood must absorb more steam before it becomes pliable, so the process takes longer. One great source for smaller bent pieces is firewood: because it is split, it usually has straight grain, and it is air dried.
A typical steam box is a simple plywood construction sized to match the material you intend to bend. It needs a door to keep the steam in and get the parts in and out, an inlet for the steam, a drain for condensation, and risers allow the heat to get to all sides of the workpiece. To generate the steam, you can use a wallpaper steamer, clothes steamer, or even a large metal tea pot. Note: steam is dangerous! Wear heavy work gloves when handling hot parts and avoid direct contact with the steam.
Getting the desired shape can be as simple as freehand bending (this works best with thinner, longer pieces), or as complicated as a multi-part jig with compression straps, custom wedges, and dozens of clamps. After it's released from the form, bent wood tends to spring back a little, so it's a good idea to over-bend slightly.
Once you’ve built your steam box and hooked up the steamer, it’s important to prepare the form, assemble clamps and make all other preparations so you can work quickly when the steaming is complete.
Insert the wood in the box, seal the door and fire up the steamer. Watch the water level in the steamer and add more if necessary to ensure a steady supply of steam. As a general rule, wood needs to be exposed to steam for at least one hour for every inch of thickness, although longer is better. Some woodworkers also recommend soaking it for hours or days ahead of time.
Put on gloves, carefully remove the wood (it’ll be hot) and clamp one end to the form.
Bend the wood around the form. You may need mechanical help, such as a come-along. Add clamps as you go, spacing them as necessary so the wood conforms to its new shape. If you’re bending around a tight radius, put metal strapping on the outside of the wood before you clamp it.
Let the wood cool and dry for one to several days, then release the clamps. The wood will spring back slightly, so if you need a particular shape it’s best to over-bend it to compensate.
Take the Mystery Out of Steam-Bending by Lon Schleining in Fine Woodworking #149–May/June 2001 Issue
How to Bend Wood to Your Will by Brian Boggs in Fine Woodworking #287-Jan/Feb 2021 Issue
7 Secrets of Steam-Bending by Michael Fortune in Fine Woodworking #238–Jan/Feb 2014 Issue
Beginner’s Guide To Steam Bending Wood by Chris Deziel on Family Handyman
Wood Bending from Wood Magazine
Lon Schleining's steam box
Steam bending a blank for carving a spoon, from Fine Woodworking
Full steam bending setup from Wood Magazine