A direct modulation multi-accumulator fractional-N frequency synthesizer 1 for generating a carrier signal 150 modulated by a modulation signal 170, 121 is disclosed. The frequency synthesizer includes a Voltage Controlled Oscillator, VCO 50, having a tuning port for controlling the frequency of the signal 110 output by the VCO, a variable divider 20 and a multi-accumulator sequence generator 21 for controlling the variable divider, a reference signal generator 50, a phase detector 30 and a low pass filter 40. These elements are arranged to form a Phase Locked Loop arrangement, the directly modulated output signal of which is taken from the output of the VCO, wherein in-band modulation is performed by varying the variable divider and out-of-band modulation is performed by directly applying the modulating signal to the VCO tuning port.