  The combination and cross-correlation of the upcoming $Euclid$ data with
cosmic microwave background (CMB) measurements is a source of great expectation
since it will provide the largest lever arm of epochs, ranging from
recombination to structure formation across the entire past light cone. In this
work, we present forecasts for the joint analysis of $Euclid$ and CMB data on
the cosmological parameters of the standard cosmological model and some of its
extensions. This work expands and complements the recently published forecasts
based on $Euclid$-specific probes, namely galaxy clustering, weak lensing, and
their cross-correlation. With some assumptions on the specifications of current
and future CMB experiments, the predicted constraints are obtained from both a
standard Fisher formalism and a posterior-fitting approach based on actual CMB
data. Compared to a $Euclid$-only analysis, the addition of CMB data leads to a
substantial impact on constraints for all cosmological parameters of the
standard $\Lambda$-cold-dark-matter model, with improvements reaching up to a
factor of ten. For the parameters of extended models, which include a
redshift-dependent dark energy equation of state, non-zero curvature, and a
phenomenological modification of gravity, improvements can be of the order of
two to three, reaching higher than ten in some cases. The results highlight the
crucial importance for cosmological constraints of the combination and
cross-correlation of $Euclid$ probes with CMB data.
