Does signal ground have to be connected to actual ground?
If you have several independent circuits, for example a circuit built up on a wireless breadboard with its own power, the ground on those separate circuits needs to be connected back to the Arduino or main board whatever that might be, if you want the boards to "talk" to each other.

Assuming you are connecting some interface (e.g. UART or I2C or SPI) between the two boards, then make sure the interface includes a ground wire to tie the grounds on the boards together.  e.g. TX/RX/GND for UART, and SCL/SDA/GND for I2C, and SS/SCK/SDO/SDI/GND for SPI.   If you have multiple of these interfaces, you only need to have one ground wire between the two boards for all of them.

If you are plugging in shields to the Arduino, it will take care of connecting the grounds through the existing connectors.

The ground on the overall system does not need to be connected to earth ground (although it doesn't hurt).  Generally you will be powering your Arduino or other main board from a wall-wart, which is connected to an outlet using only two AC wires and without any third ground wire.