  The design of fixed point algorithms is at the heart of monotone operator
theory, convex analysis, and of many modern optimization problems arising in
machine learning and control. This tutorial reviews recent advances in
understanding the relationship between Demidovich conditions, one-sided
Lipschitz conditions, and contractivity theorems. We review the standard
contraction theory on Euclidean spaces as well as little-known results for
Riemannian manifolds. Special emphasis is placed on the setting of
non-Euclidean norms and the recently introduced weak pairings for the $\ell_1$
and $\ell_\infty$ norms. We highlight recent results on explicit and implicit
fixed point schemes for non-Euclidean contracting systems.
