  We modify the definition of spherical knotoids to include a framing, in
analogy to framed knots, and define a further modification that includes a
secondary 'coframing' to obtain 'biframed' knotoids. We exhibit topological
spaces whose ambient isotopy classes are in one-to-one correspondence with
framed and biframed knotoids respectively. We then show how framed and biframed
knotoids allow us to generalize quantum knot invariants to a knotoid setting,
leading to the construction of general Reshetikhin-Turaev type biframed knotoid
invariants.
