Sums of two Cantor sets arise naturally in homoclinic bifurcations, Markov and Lagrange dynamical spectra, and the spectrum of the square Fibonacci Hamiltonian. In the 1970s Palis conjectured that for generic pairs of regular Cantor sets either the sum has zero Lebesgue measure or else it contains an interval. This problem is known to be extremely difficult, and is still open for affine Cantor sets. In this talk, we will discuss the history of sums of two Cantor sets, and also introduce my recent results about sums of two homogeneous Cantor sets.