Wavelet methods give a flexible alternative to Fourier methods in non-stationary signal analysis. The concept of band-limitedness plays a fundamental role in Fourier analysis. Since wavelet theory replaces frequency with scale, a natural question is whether there exists a useful concept of scale-limitedness. Obvious definitions of scale-limitedness are too restrictive, in that there would be few or no useful scale-limited signals. This paper introduces a viable definition for scale-limited signals, and shows that the class is rich enough to include bandlimited signals, and impulse trains, among others. Moreover, for a wide choice of criteria, we show how to design the optimal wavelet for representing a given signal, and how to design robust wavelets that optimally represent certain classes of signals.