Here's one script I used for that sort of problem.

var target : Transform; //the enemy's target
var moveSpeed = 3; //move speed
var rotationSpeed = 3; //speed of turning
var attackThreshold = 1.5; // distance within which to attack
var chaseThreshold = 10; // distance within which to start chasing
var giveUpThreshold = 20; // distance beyond which AI gives up
var attackRepeatTime = 1; // delay between attacks when within range

private var chasing = false;
private var attackTime = Time.time;

var myTransform : Transform; //current transform data of this enemy

function Awake()
{
    myTransform = transform; //cache transform data for easy access/preformance 
    }


function Start()
{
     target = GameObject.FindWithTag("Player").transform; //target the player
    }

function Update () {

    // check distance to target every frame:
    var distance = (target.position - myTransform.position).magnitude;

    if (chasing) {

        //rotate to look at the player
        myTransform.rotation = Quaternion.Slerp(myTransform.rotation,
        Quaternion.LookRotation(target.position - myTransform.position), rotationSpeed*Time.deltaTime);

        //move towards the player
        myTransform.position += myTransform.forward * moveSpeed * Time.deltaTime;

        // give up, if too far away from target:
        if (distance > giveUpThreshold) {
            chasing = false;
        }

        // attack, if close enough, and if time is OK:
        if (distance < attackThreshold && Time.time > attackRepeatTime) {

                    target.SendMessage("ApplyDamage",10);
            // Attack! (call whatever attack function you like here)
            }
        if (distance < attackThreshold) {
        moveSpeed=0;
        //stop when close enough
    }

            attackTime = Time.time+ attackRepeatTime;
        }

      else {
        // not currently chasing.

        // start chasing if target comes close enough
        if (distance < chaseThreshold) {
            chasing = true;
        }
    }

}

function OnTriggerEnter (other: Collider) {
if (other.gameObject.CompareTag("Bullet")){
chaseThreshold=100000000;
}
}

it was pretty useful to get me going. should be pretty self explanatory. just comment if you have any questions

I think (just a thought) thatwhen the zombie reaches you, he reaches his destination. mission over. and then, he does nothing.

try this raycast; Physics.RayCast(transform.position,player.position,10); if (hit.distance<30) { //attack code here }

I would love some help that script works but i dont have a health script so? how do i actually get hit by the zombie? could anyone send me a health script if someone has worked that out would give you credits :)

The problem is that you set moveSpeed to 0 when the zombie reaches melee range without setting it back to 3 when you are away again.