Discrete parts of gauge redundancies and certain discrete global symmetries are also known to acquire anomalous phases. This chapter is devoted to such discrete anomalies that involve the eta invariant and the Atiyah-Patodi-Singer index theorem. After a brief derivation of the latter theorem and exploration of the eta invariant, we turn to the parity anomaly of odd-dimensional fermions. Particular attention is given to how the eta invariant is the true quantity that captures the parity anomaly, rather than the Chern-Simons effective action; the latter is merely a continuously variable part of the former and has led to various confusions over fermions in odd dimensions. The same mathematical devices for Weyl fermions in even dimensions result in the celebrated Witten anomaly, say, for d=4 Sp(k) theories, which is given a full treatment. The final part of the chapter is devoted to a relatively recent and extremely versatile model by Witten and Yonekura that combines these two discrete anomalies, despite their very distinct physical consequences, into a single framework that realizes the fermions in question as edge modes of infinitely massive bulk Dirac fermions.