Novel analytic bond-order potentials (BOP's) are derived for the sigma and pi bonds of sp-valent systems that are correctly bounded from above by unity. We show that these BOP's allow the concept of single, double, triple, and conjugate bonds in carbon systems to be quantified, the average error compared to accurate tight-binding predictions being only 1% for the sigma bonds and 15% for the pi bonds. Although molecular dynamics simulations are an order of magnitude slower than with standard Tersoff potentials, these new BOP's provide the first "classical" interatomic potentials that handle both structural differentiation and radical formation naturally within its framework.