I'm not sure how many the UNO can handle natively but I use the Adafruit 16-Channel 12-bit PWM/Servo Shield - I2C interface which allows 16 servos. Using the shield allows the UNO to perform more functions without having to manage servos.
The UNO has 5 pins which are PWM capable. Pins 5,6,9.10 and 11, or you can use the board suggested above.