Variety of climbing shoes.
Climbing shoes have special rubber soles that increase the
friction coefficient. Climbing shoes have rubber of all kinds,
and climber chose them based off what they want. The lower the
friction coefficient of the rock the higher the friction
coefficient of the shoes needs to be. Smooth rock with require
soft rubber, and rough rock will use harder rubber.
Climbers often chose to climb on dry days
because water in the air, or on rock will decrease the
friction coefficient. Even with soft rubber, a climber will
still slip easier because the rubber is smooth until disturbed
by rough rock.
Climbers will also use chalk. Chalk can be
used on sweaty hands to increase the friction, and allow the
climber to continue up the wall.
Climbing is often described as, "a battle
between gravity and friction".
Photo from: http://survival-mastery.com/reviews/rock-climbing-gear.html