Medicine ball core exercises are great because they add force of momentum to your core-conditioning program. Your core muscles has to produce force to get the medicine ball to move and to stop. This helps improve your results over time. Below is a list of core exercises you can do with a medicine ball. 
Primary Muscle Group: Abdominal Obliques
Muscle Groups Worked in This Exercise: Obliques and Rectus Abdominis
Preparation: Stand erect with back and head straight up. Medicine ball should be placed between hands and directly overhead with elbows just slightly bent.
Breathing: Breath out on the contraction of the muscle and breath back in on the way back to the start position.
Primary Muscle Group: Abdominal Obliques
Muscle Groups Worked in This Exercise: Obliques and Rectus Abdominis
Preparation: Hook feet under foot brace and lie supine on incline board with hips bent. Hold medicine ball close to chest with both hands.
Breathing: Breath out on the contraction of the muscle and breath back in on the way back to the start position.