Most woodworkers today make wooden bowls using a lathe, but you can also make them by hand using only simple tools like a gouge, chisel and handsaw. You won't need to rely on electricity or a lathe.
To carve a durable round bowl that's less likely to split or warp, choose a tight-grained hardwood, like beech, elm, birch or cherry, though you also can use softwoods. The most stable woods for bowls come from a part of the tree with a random grain, such as a burl or the root ball.
Saw a block of wood as deep as the bowl you'll be carving and 4 or 5 inches wider all around. The top and bottom sides should be parallel to each other but otherwise the shape doesn't matter at this point. It can be square, oblong or uneven, as long as you have several inches of extra wood beyond the circle of the bowl. Cut the wood so the grain is running across the bowl and not in concentric rings centred at the bottom of the bowl, to prevent splitting or cracking. If the wood isn't kiln dried, let it season thoroughly in a dry place. Seasoning may take several weeks or months if you cut the wood while it was green but making sure the wood is thoroughly dry helps prevent your bowl from warping after you carve it.
Clamp the block of wood to a worktable, right side up, placing the clamp in the waste area of the wood outside where the final bowl will be. Mark the centre of the bowl and use a compass to draw a circle on the top of the wood, outlining the inside edge of the bowl. Place a ruler against the centre mark and draw several lines radiating out from the centre beyond the edge of the bowl. You can use these to align a cardboard pattern to check the depth of the bowl.
Cut a cardboard pattern to help you gauge your progress while carving the inside of the bowl. Make it as wide and deep as the inside of the bowl and curved like you want the sides to be.
Begin carving away the wood to create the inside of the bowl, using a gouge or chisel. Check your work with the cardboard pattern, lining it up so that the edges rest against the lines your drew, so it always runs through the centre of the bowl. Work slowly as you get near the finished depth and be careful not to cut through what will be the wall of the bowl. Finish the rough shape of the inside, though you don't need to give it a final sanding yet.
Draw a circle to mark the outside top of the bowl. Saw straight down on four sides to make a square just that size, then saw off the corners to make an octagon. Saw off the lower corners to rough-in the angled sides of the bowl. Use a chisel to finish shaping the outside. Cut another cardboard pattern to fit the outside of the bowl, to help you gauge how much to cut away. Work carefully as you get near the end, so you don't remove too much.
Smooth the inside and outside of the bowl with sandpaper, starting with a coarse grit and working to a finer one until you've polished the bowl to your satisfaction. If you'll use the bowl for food, finish it with a food-grade oil such as salad bowl oil or mineral oil. If you won't be putting food in it, you can use any wood finish.