We think puzzles are for re-doing, but some of you big-game-hunters out there like to mount your puzzles, and have asked what's the best way to mount a wooden jigsaw puzzle. We believe a standard wood glue (available at your local hardware store should do it).
Customer Christine Z in PCB has another suggestion: just press your finished puzzle down on Post-it Self-stick Bulletin Board. This is very fast and simple. You can cut this bulletin board to size, and it costs around $15 on amazon.com (and possibly at your local office supplies store).
Let us know if you try it or have another idea! (email@example.com).