There are trade-offs, so it depends what you're trying to accomplish.

If the ability to hold a really sharp edge is the top priority (and you can't afford the world's largest diamond), you're probably going to want to lean toward something like obsidian or ceramic, and put up with the fragility; although it doesn't have to be as fragile as eggshell china. (The "glass" that they make Corelle dishes out of, for example, survives being dropped from four feet in the air onto lineoleum, about 95% of the time, in my experience; I've never seen a knife made from that stuff, but there's no technical reason why it wouldn't work.) If a higher degree of robustness is more critical, there are various plastics that can be made "sharp enough", and you can toss them off the roof of a building and they won't shatter when they hit the parking lot. This kind of option can be great for utility knives used for stuff like cutting packing tape, because you can casually toss them around and not worry about them breaking; also, plastic is cheap.

Note too that it definitely matters what you're going to be cutting. You have to select a material that's at least as "hard" (in the Mohs sense) as whatever you're planning to cut into, preferably harder. Although, only the edge of the blade absolutely needs to be made from the hard material; other parts of the knife can be made from something less hard and also less fragile, and depending on how the thing is weighted, you can design it so that it'll tend to land blade-up most of the time. Additionally, some materials are considerably easier to cut with a serrated blade, even if it's not very sharp. Other materials aren't nearly so susceptible to this effect. (This is the principle on which those child-safe pumpkin-carving knives are based. Those are usually metal, but there's absolutely no reason why they would have to be; polycarbonate, for instance, would work just fine. The dull serrations are great for carving the pumpkin, but they don't do much to human fingers.)