The zero modes and zero resonances of the Dirac operator H = α D + Q(x) are discussed, where α = (α1, α2, α3) is the triple of 4 × 4 Dirac matrices, D = (1/i)∇x, and Q(x) = qjk(x))is a 4 × 4 Hermitian matrix-valued function with |qjk(x)| ≤ C(x)-ρ, ρ > 1. We shall show that every zero mode f(x) is continuous on R3 and decays at infinity with the decay rate |x|-2. Also, we shall show that H has no zero resonance if ρ > 3/2. © 2008 by the University of Notre Dame. All rights reserved.