  We present the first comprehensive study of hybrid baryons using lattice QCD
methods. Using a large basis of composite QCD interpolating fields we extract
an extensive spectrum of baryon states and isolate those of hybrid character
using their relatively large overlap onto operators which sample gluonic
excitations. We consider the spectrum of Nucleon and Delta states at several
quark masses finding a set of positive parity hybrid baryons with quantum
numbers N_1/2+, N_1/2+, N_3/2+, N_3/2+, N_5/2+ and Delta_1/2+, Delta_3/2+ at an
energy scale above the first band of `conventional' excited positive parity
baryons. This pattern of states is compatible with a color octet gluonic
excitation having J^P=1+ as previously reported in the hybrid meson sector and
with a comparable energy scale for the excitation, suggesting a common
bound-state construction for hybrid mesons and baryons.
