  We develop a new embedding-space formalism for AdS$_4$ and CFT$_3$ that is
useful for evaluating Witten diagrams for operators with spin. The basic
variables are Killing spinors for the bulk AdS$_4$ and conformal Killing
spinors for the boundary CFT$_3$. The more conventional embedding space
coordinates $X^I$ for the bulk and $P^I$ for the boundary are bilinears in
these new variables. We write a simple compact form for the general
bulk-boundary propagator, and, for boundary operators of spin $\ell \geq 1$, we
determine its conservation properties at the unitarity bound. In our CFT$_3$
formalism, we identify an $\mathfrak{so}(5,5)$ Lie algebra of differential
operators that includes the basic weight-shifting operators. These operators,
together with a set of differential operators in AdS$_4$, can be used to relate
Witten diagrams with spinning external legs to Witten diagrams with only scalar
external legs. We provide several applications that include Compton scattering
and the evaluation of an $R^4$ contact interaction in AdS$_4$. Finally, we
derive bispinor formulas for the bulk-to-bulk propagators of massive spinor and
vector gauge fields and evaluate a diagram with spinor exchange.
