While the use of sinusoidal electromagnetic waves as information carriers is taken as one of the principal axioms of today's wireless system design, certain nonsinusoidal waves may further enrich the scope and capacity of modern wireless engineering. Two notable nonsinusoid examples are impulses and chaotic signals. The short temporal width of impulses has enabled applications such as ranging radars and ultra wideband (UWB). The complex nature of chaotic signals offers a new means of encrypted communication. Here we review a new circuit paradigm, the electrical soliton oscillator, which can self-generate both impulse and chaotic signals of very large bandwidth by leveraging the singular dynamics of a nonlinear wave known as the electrical soliton. By combining a nonlinear transmission line with a unique amplifier that can "tame" the inherently unruly dynamics of solitons, the oscillator self-generates a stable, periodic train of short impulses. If the taming functions of the amplifier are turned off, the circuit self-generates chaotic signals by positively exploiting solitons' unruly nature. While still in its early stages, this soliton circuit may one day serve as the heartbeat of both impulse and chaotic wireless systems