The security of essentially all public-key cryptography currently in common use today is based on the presumed computational hardness of three number-theoretic problems: integer factoring (required for the security of RSA encryption and digital signatures), discrete logarithms in finite fields (required for Diffie-Hellman key exchange and the DSA digital signature algorithm), and discrete logarithms over elliptic curves (required for elliptic curve Diffie-Hellman and ECDSA, Ed25519, and other elliptic curve digital signature algorithms). In this column, we will review the current state of the art of cryptanalysis for these problems using classical (non-quantum) computers, including in particular our most recent computational records for integer factoring and prime field discrete logarithms.