  We develop Weiss's manifold calculus in the setting of $\infty$-categories,
where we allow the target $\infty$-category to be any $\infty$-category with
small limits. We will establish the connection between polynomial functors, Kan
extensions, and Weiss sheaves, and will classify homogeneous functors. We will
also generalize Weiss and Boavida de Brito's theorem to functors taking values
in arbitrary $\infty$-categories with small limits.
