Cambridge University Press link: Cambridge. Amazon link: Amazon.
Now translated into Japanese by Ryuhei Uehara: http://www.jaist.ac.jp/~uehara/books/howtofoldit/ :
[Web site always under construction! Last updates: 29 July 2013.]
The book includes descriptions of ten open problems (see the Index under open problem, p. 176). Inevitably, some will be resolved as time passes and before a Second Edition is prepared. Here I will maintain a list of those Open Problems in the book which have been closed, or on which some notable advance has been achieved.
The special case of convex polyhedra has been solved (Yes, they can all be continuously flattened!) by Jinichi Itoh, Chie Nara, and Costin Vilcu in the following paper: "Continuous flattening of convex polyhedra," XIV Spanish Meeting on Computational Geometry, 2011, Centre de Recerca Matematica Documents Vol. 8, pp. 9598.
See also the longer list of Closed/Open Problems associated with the more advanced monograph, Geometric Folding Algorithms: Linkages, Origami, Polyhedra.
Below appears some supplementary material, organized by chapter. Click on links to jump to that chapter's material:
Animation by Akira Nishihara. (Click on image to see animated GIF): Nishihara web page.  

Fig. 3.14 (p.51)  Uncurling of a unit 90°chain as it stretches to its maxspan configuration. [Video by Julia Patterson] 
Final frame of animation 
The video is a first attempt at animating the uncurling of the spiral unit 90°chain. The jittering near the end of the stretch is under investigation...
Click on thumbnail images to bring up 8½ x 11" PDF ready for printing.
Where  Description  Templates 

Fig. 4.4  Degree6 vertex  
Fig. 4.7  Degree6 nonflat  
Fig. 4.8  MaekawaJustin Theorem  
Fig. 4.14  Map puzzle 
Click on thumbnail images to bring up 8½ x 11" PDF ready for printing.
Where  Description  Templates  

Fig. 5.2  Square  
Fig. 5.3  Star  
Fig. 5.5  Rectangle  
Fig. 5.7  Irregular Triangle  
Fig. 5.9  Square & Rectangle  
Fig. 5.11  Letter 'A' [This corrects one incorrectly labeled crease in Fig.5.11(c).] 

Letter 'A' with hole  
Fig. 5.12  Turtle  
Here is a dynamic version of Figure 7.11: The Latin cross unfolding of the cube.
Cube to Latin cross. [Video by Katie Park] 

Here is an edgeunfolding of a cube to a 'Z'shape, and then a refolding of the 'Z' to doublecovered parallelogram. (This example is not in the book, but related.) All the Platonic solids but the dodecahedron have similar edgeunfoldings and refoldings to a parallelogram. Template below in Chapter 9 section.
Unfolding cube to 'Z'.  Folding 'Z' to parallelogram. 

[Videos by Katie Park] 
These four videos show animations of the orthogonal terrain algorithm described in Section 8.1, especially Figures 8.2 and 8.4. For more detail, see my note, "Unfolding Orthogonal Terrains."
10 x 10 Terrain, Example 1  10 x 10 Terrain, Example 2 

10 x 10 Top only (dynamically rescaled)  20 x 20 Terrain 
The video below illustrates the basic construction that underlies the algorithm described in Section 8.4 Above & Beyond, especially Figure 8.8. For more detail, see the paper, "Epsilon Unfolding of Orthogonal Polyhedra."
Helical unwrapping one box to a staircase shape. [Video by Robin Flatland and Ray Navarette] 

Where  Description  Templates 

(Not in book) 
Which folds to cube? 

(Not in book) 
Cube Z → Octahedron 

Fig. 9.6  Square perimeterhalving.  
(Not in book. 
Cube Z to Parallelogram  
Fig. 9.9  Latin cross to Tetrahedron  
Fig. 9.10  'SoCG' polygon: Folds to what? 