Make a camera look at an object, make it smoothly follow its path, and don't jerk-turn around when it moves beyond 0 or 360 degrees, dammit.

CameraLookAt.cs

using UnityEngine;
using System.Collections;

/// <summary>
/// Put this script on a camera, assign a target and the camera will always look at that target, smoothly following it.
/// </summary>
public class CameraLookAt : MonoBehaviour {

	public Transform target; //Look at this
	private Transform aimTarget;
	private Vector3 aimAngle = Vector3.zero; //eulerAngles from aimTarget will go in here
	private Vector3 finalAngle = Vector3.zero; //this is not clamped between 0 and 360, it will go on and on... 

	void Start() {
		//Ok. We can't do LookAt directly on this transform, because we want smooth motion.
		//So, let's create a Transform that will do the LookAt, then we'll lerp towards it's eulerAngles
		GameObject go = new GameObject("CameraLookAt AimTarget");
		go.transform.parent = transform.parent;
		go.transform.position = transform.position;
		aimTarget = go.transform;
		aimAngle = finalAngle = transform.eulerAngles;
	}


	void LateUpdate () {
		aimTarget.LookAt(target, Vector3.up); //dummy looks at target
		aimAngle = aimTarget.eulerAngles; //take angles from dummy

		//this is where the magic happens. let's add or subtract 360f until aimAngle is within 360f of the finalAngle
		while(aimAngle.y - finalAngle.y > 180f) aimAngle.y -= 360f; 
		while(aimAngle.y - finalAngle.y < -180f) aimAngle.y += 360f;

		finalAngle = Vector3.Lerp(finalAngle, aimAngle, Time.deltaTime * 2f); //now lerp towards the aligned aimAngle.

		//this might come as a surprise. after all the hassle to get around the 0-360 clamping that happens in eulerAngles, we're now clamping it back to 0-360. 
		while(finalAngle.y < 0f) finalAngle.y += 360f; //why? to avoid errors due to float imprecision.
		while(finalAngle.y > 360f) finalAngle.y -= 360f; //uncomment these 2 lines, turn the camera around its axis 1000 times and you'll see what I mean. (it will start shaking/studdering)

		transform.eulerAngles = finalAngle; //assign finalAngle, done!
	}

	/// <summary>
	/// Assign a new target to look at
	/// </summary>
	/// <param name="t">the new target</param>
	public void LookAtMe(Transform t) {
		target = t;
	}
}