  Galaxy cluster counts in bins of mass and redshift have been shown to be a
competitive probe to test cosmological models. This method requires an
efficient blind detection of clusters from surveys with a well-known selection
function and robust mass estimates. The Euclid wide survey will cover 15000
deg$^2$ of the sky in the optical and near-infrared bands, down to magnitude 24
in the $H$-band. The resulting data will make it possible to detect a large
number of galaxy clusters spanning a wide-range of masses up to redshift $\sim
2$. This paper presents the final results of the Euclid Cluster Finder
Challenge (CFC). The objective of these challenges was to select the cluster
detection algorithms that best meet the requirements of the Euclid mission. The
final CFC included six independent detection algorithms, based on different
techniques, such as photometric redshift tomography, optimal filtering,
hierarchical approach, wavelet and friend-of-friends algorithms. These
algorithms were blindly applied to a mock galaxy catalog with representative
Euclid-like properties. The relative performance of the algorithms was assessed
by matching the resulting detections to known clusters in the simulations.
Several matching procedures were tested, thus making it possible to estimate
the associated systematic effects on completeness to $<3$%. All the tested
algorithms are very competitive in terms of performance, with three of them
reaching $>80$% completeness for a mean purity of 80% down to masses of
$10^{14}$ M$_{\odot}$ and up to redshift $z=2$. Based on these results, two
algorithms were selected to be implemented in the Euclid pipeline, the AMICO
code, based on matched filtering, and the PZWav code, based on an adaptive
wavelet approach. [abridged]
