A candle has a wax or wax-simulating body with a top wall, a simulated burnt wick, a cylindrical side wall, and an interior candle cavity. A partly transparent housing with top and side walls together defining a housing cavity, is located in the candle cavity. An annular, transparent flange extends upwardly to a lower surface of the top wall of the body. An illumination assembly connected to the housing, extends into the housing cavity and includes an LED, a circuit for powering the LED and a battery compartment or other power source for powering the circuit. The lamp is spaced from the top wall of the housing and positioned for casting light through the top wall of the housing, into a light directing space defined by the flange, and to the top wall of the candle body.