GCD sums naturally appear in a Diophantine approximation problem considered by Hardy and Littlewood in 1922. It was until 1949 that Gál solved their problem. Since Gál's fundamental theorem, there have been new developments in the past decade. I will review these results and report some new results on the limit of GCD sums and the log-type GCD sums. Another topic of the talk concerns large values of derivatives of zeta and L-functions. Large values of zeta and L-functions are classical topics in analytic number theory, which can be dated back to a result of Bohr and Landau in 1910. Resonance methods are modern tools to produce large values of zeta and L-functions. GCD sums are one of important ingredients. I will talk on producing large values of derivatives of zeta and L-functions via resonance methods. The link between log-type GCD sums and derivatives of the Riemann zeta function will also be discussed.