In this survey of recent computational and theoretical work I will report on the following interrelated topics:

• (1) computational efforts to find new Wieferich and Wilson primes,
• (2) theoretical and computational results on Fermat and Wilson quotients for composite moduli,
• (3) the search for new factors of Fermat numbers, and
• (4) factors of generalized Fermat numbers.

The talk is partly based on various joint work with T. Agoh, R. Brent, R. Crandall, C. van Halewyn, C. Pomerance, and L. Skula.

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