Animate a view zoom-bouncing in?

Too many complicated answers 😊. It's much easier to do it as of 2017 (Swift 3/4).

Swift 4-

yourview.transform = yourview.transform.scaledBy(x: 0.001, y: 0.001)

UIView.animate(withDuration: 0.5, delay: 0.0, usingSpringWithDamping: 0.5, initialSpringVelocity: 0.3, options: .curveEaseInOut, animations: {
      yourview.transform = CGAffineTransform.identity.scaledBy(x: 1.0, y: 1.0)
  }, completion: nil)

Note the usingSpringWithDamping parameter. This parameter dictates how much bounce/spring effect you want and allows values from 0 to 1. The lower the value the more the bounce effect. Also, if you do not like the scaleBy effect then you can simply use good old frames to show a expanding and bouncing UIView.

Just refactored the code by Amit Singh using blocks, which makes it much simpler and readable.

popUp.transform = CGAffineTransformScale(CGAffineTransformIdentity, 0.001, 0.001);

[self.view addSubview:popUp];

[UIView animateWithDuration:0.3/1.5 animations:^{
    popUp.transform = CGAffineTransformScale(CGAffineTransformIdentity, 1.1, 1.1);
} completion:^(BOOL finished) {
    [UIView animateWithDuration:0.3/2 animations:^{
        popUp.transform = CGAffineTransformScale(CGAffineTransformIdentity, 0.9, 0.9);
    } completion:^(BOOL finished) {
        [UIView animateWithDuration:0.3/2 animations:^{
            popUp.transform = CGAffineTransformIdentity;                            
        }];
    }];
}];

This can be done in a much simpler way these days:

view.transform = CGAffineTransformScale(CGAffineTransformIdentity, 0.001, 0.001);
[UIView animateWithDuration:0.5 delay:0.2 usingSpringWithDamping:0.6f initialSpringVelocity:1.f options:UIViewAnimationOptionCurveEaseInOut animations:^{
    view.transform = CGAffineTransformScale(CGAffineTransformIdentity, 1.0, 1.0);
} completion:^(BOOL finished) {
    view.transform = CGAffineTransformIdentity;
}];

write this code when you want to trigger this animation

popUp.transform = CGAffineTransformScale(CGAffineTransformIdentity, 0.001, 0.001);

[self.view addSubview:popUp];

[UIView animateWithDuration:0.3/1.5 animations:^{
    popUp.transform = CGAffineTransformScale(CGAffineTransformIdentity, 1.1, 1.1);
} completion:^(BOOL finished) {
    [UIView animateWithDuration:0.3/2 animations:^{
        popUp.transform = CGAffineTransformScale(CGAffineTransformIdentity, 0.9, 0.9);
    } completion:^(BOOL finished) {
        [UIView animateWithDuration:0.3/2 animations:^{
            popUp.transform = CGAffineTransformIdentity;                            
        }];
    }];
}];

SWIFT 5.0

    selectView.transform =
        CGAffineTransform.identity.scaledBy(x: 0.001, y: 0.001)

    view.addSubview(selectView)

    UIView.animate(withDuration: 0.3 / 1.5, animations: {
        selectView.transform =
            CGAffineTransform.identity.scaledBy(x: 1.1, y: 1.1)
    }) { finished in
        UIView.animate(withDuration: 0.3 / 2, animations: {
            selectView.transform = .identity.scaledBy(x: 0.9, y: 0.9)
        }) { finished in
            UIView.animate(withDuration: 0.3 / 2, animations: {
                selectView.transform = CGAffineTransform.identity
            })
        }
    }

This is updated code (from fabio.cionini) as it is accepted answer so updating to latest.