Conformal field theory (CFT) provides a very powerful framework to study the large-scale properties of models of statistical mechanics at their critical point. The prototypical example of this is the continuum (scaling) limit of the two-dimensional critical Ising model. The case of critical percolation is more difficult, partly because its continuum limit is believed to be described by a relatively unusual type of CFT, called a logarithmic CFT. In this talk, I will first briefly explain the statements above. I will then present some recent results and work in progress that are part of a program aimed at fitting percolation within the logarithmic CFT framework.