When I got a dog, I realized the cats were spending a lot less time with me in the living room. I decided to build them an escape so they could stick around and stay out of the puppy's reach.

Here is how I built a curved staircase to ensure safe passage from the couch to the top of a bookcase.

Step 1: Stringer Molding

Mold the outer stringer which will define your stair case's curve by soaking wood in water and clamping to a form.

Cut a sheet of plywood into two 4'x4' pieces
draw a circle onto each piece that will be the diameter of your staircase's curve
screw studs connecting the two pieces of plywood (kind of like a half-pipe)
mark a length of about 4' (the eventual height of your staircase) on the first and last stud
Soak lengths of 1/8" by 4" hard wood in water (a kiddie pool) for a few hours
clamp 1/8" strips to the frame (layer two thick and make sure to stagger them if they aren't long enough to span the whole length), curving from your start mark on the first stud to the stop mark on the last one.
Allow to dry
Unclamp, glue the two thicknesses of 1/8" together, and reclamp.
Allow to dry

Step 2: The Steps

Use your stringer form to create a stair step template

Remove the studs from your stringer form
Draw a circle 6" inside one of the existing circles; this separation will be the width of your step.
Do some math: you'll want to make 90 deg worth of steps, so figure out how deep you want each step and how high each step should be. (I did about 10 steps, 4" deep on the inner circle and 4" tall)
Cut out one step to use as a template. Also cut a template for your first and last step which follows the inner circle but not the outer: you want this step to lay flat on the wall rather then curving back in
Rough cut some steps (I used 1/2" hard wood) and trace your template with a router for some smooth curves
Also cut out the upright pieces: a bunch of rectangles (4" x 6" x 1/8" in my case)
Glue these all together. Either take a bunch of time or make a sculpture with every clamp you own like I did. You might want some screws too (drill pilot holes!)

Step 3: Assembly

Now it's starting to come together: All that's left is connecting your stairs to the stringer and painting

Glue the stringer to your steps. My stringer didn't totally cover the stairs rise/run, so I lined it up on the top and the overlap underneath isn't really noticeable.
Tack the stringer in place with some small screws. Drill pilot holes! it would be a shame to split anything at this stage.
Paint it your (or your cat's) favorite color.

n.b. The stringer will likely not make a perfect 90deg and be a little slack. This is ok, thats why we're supplementing our glue with screws.