  We initiate a systematic study of 3-dimensional `defect' topological quantum
field theories, that we introduce as symmetric monoidal functors on stratified
and decorated bordisms. For every such functor we construct a tricategory with
duals, which is the natural categorification of a pivotal bicategory. This
captures the algebraic essence of defect TQFTs, and it gives precise meaning to
the fusion of line and surface defects as well as their duality operations. As
examples, we discuss how Reshetikhin-Turaev and Turaev-Viro theories embed into
our framework, and how they can be extended to defect TQFTs.
