  The radio, optical, and $\gamma$-ray light curves of the blazar S5 1803+784,
from the beginning of the {\it Fermi} Large Area Telescope (LAT) mission in
August 2008 until December 2018, are presented. The aim of this work is to look
for correlations among different wavelengths useful for further theoretical
studies. We analyzed all the data collected by {\it Fermi} LAT for this source,
taking into account the presence of nearby sources, and we collected optical
data from our own observations and public archive data to build the most
complete optical and $\gamma$-ray light curve possible. Several $\gamma$-ray
flares ($\mathrm{F>2.3~10^{-7} ph(E>0.1 GeV)~cm^{-2}~s^{-1}}$) with optical
coverage were detected, all but one with corresponding optical enhancement; we
also found two optical flares without a $\gamma$-ray counterpart. We obtained
two {\it Swift} Target of Opportunity observations during the strong flare of
2015. Radio observations performed with VLBA and EVN through our proposals in
the years 2016-2020 were analyzed to search for morphological changes after the
major flares. The optical/$\gamma$-ray flux ratio at the flare peak varied for
each flare. Very minor optical V-I color changes were detected during the
flares. The X-ray spectrum was well fitted by a power law with photon spectral
index $\alpha$=1.5, nearly independent of the flux level: no clear correlation
with the optical or the $\gamma$-ray emission was found. The $\gamma$-ray
spectral shape was well fitted by a power law with average photon index
$\alpha$= 2.2. These findings support an Inverse Compton origin for the
high-energy emission of the source, nearly co-spatial with the optically
emitting region. The radio maps showed two new components originating from the
core and moving outwards, with ejection epochs compatible with the dates of the
two largest $\gamma$-ray flares.
