I see no shadows at all!

• Shadows are a Unity Pro only feature, so without Unity Pro you won't get shadows. Simpler shadow methods, like using a Projector, are still possible of course.
• Shadows also require certain graphics hardware support. See Shadows page for details.
• Check if shadows are not completely disabled in Quality Settings.
• Shadows are currently not supported for iOS and Android mobile platforms.

Some of my objects do not cast or receive shadows

First, the Renderer has to have Receive Shadows on to have shadows on itself; and Cast Shadows on to cast shadows on other objects (both are on by default).

Next, only opaque objects cast and receive shadows; that means if you use built-in Transparent or Particle shaders then you'll get no shadows. In most cases it's possible to Transparent Cutout shaders (for objects like fences, vegetation etc.). If you use custom written Shaders, they have to be pixel-lit and use Geometry render queue. Objects using VertexLit shaders do not receive shadows either (but can cast shadows just fine).

Finally, in Forward rendering path, only the brightest directional light can cast shadows. If you want to have many shadow casting lights, you need to use Deferred Lighting rendering path.