An alternative is to use a component that generates signals directly. Analog Devices have several such. They are known as DDS devices, for direct digital synthesis. The idea is that the DDS generates a digital value internally which it then converts to an analog output. The disadvantage of such devices is that they can't generate arbitrary waveforms. For example, the AD9833 generates sine, triangle and square waves, and the AD9850 generates only sine and square waves. They are not really intended for arbitrary signal generation, and instead are aimed more at things such as modulating and demodulating signals.
I put together a signal generator using an Arduino, a prebuilt AD9833 module and some components to provide a user interface: a LCD display, and a rotary encoder. You can pick up AD9833 modules quite cheaply on ebay. AD9850 modules are more widely available and a bit cheaper. Here is the wiring diagram for how I connected it up:
There is code for it here. Roughly, the idea is to show three fields in the display: the frequency (1Hz to 1MHz), the waveform and whether automatic or manual mode is in use. Automatic mode means changes take effect immediately. Pressing the switch (actually the shaft of the rotary encoder) has various effects. A long press moves between the three fields, indicated by an arrow next to the active field. While in the frequency field, turning the encoder increases or decreases the frequency, and a short press of the button select which digit you are altering. For example, if the current frequency is 10000Hz and the current digit, indicated by a cursor, is the middle one, the frequency changes in steps of 100Hz. On the waveform field, the encoder steps between sine, square and triangle. On the automatic/manual field, the encoder steps between these options, and a short press on the button while in manual mode applies the current settings.
Originally I had thought of making this a full project, packaging it up and adding more options to the software. For now, I am leaving it like this, as I have some ideas for a different signal generator project that I would like to explore.