  We study on-shell scattering amplitudes for continuous-spin particles.
Poincar\'e invariance, little-group $ISO(2)$ covariance, analyticity, and
on-shell factorization (unitarity) impose stringent conditions on these
amplitudes. We solve them by realizing a non-trivial representation for all
little-group generators on the space of functions of bi-spinors. The
three-point amplitudes are uniquely determined by matching their high-energy
limit to that of definite-helicity (ordinary) massless particles. Four-point
amplitudes are then bootstrapped using consistency conditions, allowing us to
analyze the theory in a very transparent way, without relying on any off-shell
Lagrangian formulation. We present several examples that highlight the main
features of the resulting scattering amplitudes. Finally, we explore under
which conditions it is possible to relax some assumptions, such as strict
on-shell factorization, analyticity, or others. We show that continuous-spin
particle dynamics may approximate gravity and electromagnetism in a loose
version of $S$-matrix principles.
