Three.js: Get the Direction in which the Camera is Looking

The camera is looking down its internal negative z-axis. So create a vector pointing down the negative z-axis:

var vector = new THREE.Vector3( 0, 0, - 1 );

Now, apply the same rotation to the vector that is applied to the camera:

vector.applyQuaternion( camera.quaternion );

You can get the angle in radians to the target like so:

angle = vector.angleTo( target.position );

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EDIT: You can now get the direction in which the camera is looking like so:

var vector = new THREE.Vector3(); // create once and reuse it!
...
camera.getWorldDirection( vector );

Note: By passing in the vector in which to store the result, the method will not have to instantiate a new THREE.Vector3 every time the method is called.

Updated to three.js r.107

fluffybunny nailed it. With a small change to his awesome idea, you can get the rotation in degrees:

var vector = camera.getWorldDirection();
angle = THREE.Math.radToDeg( Math.atan2(vector.x,vector.z) );  

I spent a long time trying to figure out captain obvious's question.

This is how it finally worked for me:

    vector = camera.getWorldDirection();
    theta = Math.atan2(vector.x,vector.z);

theta is in radians. This is how I orient my game's character to face the same way the camera is facing. The getWorldDirection() is a fairly new option on camera.