**Abstract**

The Calderón operator is the sum of the Hardy averaging operator and its adjoint, and plays an important role in the theory of real interpolation. On the other hand, the Hilbert operator arises from the continuous version of Hilbert's inequality. Both operators appear in different contexts and have numerous applications within harmonic analysis. In this chapter we will briefly review the Calderón and Hilbert operators, showing some of the most relevant results within functional analysis and finally we will present recent results on these operators within Fourier analysis.

**Keywords:** Calderón operator, Hilbert operator, Lebesgue spaces, Lipschitz spaces, BMO spaces, weighted inequalities, Calderón weights
