Slide show of Core Exercises (no equipment):
http://www.mayoclinic.com/health/core-strength/SM00047
Slide show of Core Exercises (with stability ball):
http://www.mayoclinic.com/health/core-strength/SM00046
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Core Exercises on the Ball
The exercise ball is a great tool for strengthening the abs, but it also helps you reach those hard-to-get-to muscles (such as the TVA and erector spinae), as well as improving your balance and overall coordination. The following exercises offer challenging ways to work a variety of muscle groups. Many of these moves are advanced exercises that require previous experience with an exercise ball. If you're new to the exercise ball, try this Beginner Ball Workout.
- When first trying these exercises, you may want to hold onto a wall or prop the ball against something sturdy for added stability.
- Perform 1-3 sets of 10-16 reps of each exercise, taking care to perfect your form before increasing sets or reps.
- Avoid any moves that cause pain or that you're not clear how to do correctly.
- Always see your doctor before exercising if you have any existing injuries or conditions.
Back Extension 
Position the ball under your hips and lower torso with the knees straight or bent. With hands behind the head or behind back, slowly roll down the ball. Lift your chest off the ball, bringing your shoulders up until your body is in a straight line. Make sure your body is in alignment (i.e., head, neck, shoulders and back are in a straight line), your abs are pulled in and that don't hyperextend the back.
Ball Balance 
Position the ball under your abs and hips, hands on the floor and legs straight and off the floor. Hold that position for 20 to 30 seconds, keeping your body in a straight line, abs pulled in. Keeping balance, slowly raise your right arm out to the side, taking care not to roll or allow any part of your body to collapse. Hold that for a few seconds and switch arms. This is tougher than it looks!