[paulat (OP)]

How many atoms are there in the Universe? And how is such a calculation considered?

[chiralSPO]

Unfortunately, we don't know how big the universe is. There doesn't appear to be any boundary, and it appears to be flat, so it is possible that the universe is infinite, in which case there would be infinitely many atoms (and infinitely many stars).

From what we can see (the observable universe; https://en.wikipedia.org/wiki/Observable_universe), there appears to be 10⁵³ kg of ordinary matter, most of which is hydrogen. If we assume that it is all hydrogen, then there would be about 10⁷⁹ atoms. This is almost certainly more atoms than there actually are, but serves as a reasonable upper limit.

Questions of mass aside, there is also a question of whether one could actually count the matter in stars as atoms. Sure there are protons and electrons in almost exactly a 1:1 ratio, but it's too hot for them to actually settle into being atoms (it is a plasma https://en.wikipedia.org/wiki/Plasma_(physics))

[apaparg]

The following info might be outdated but on late 30s, Sir Arthur Stanley Eddington claimed that "I believe there are
15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,
914,527,116,709,366,231,025,076,185,631,031,296
protons in the universe, and the same number of electrons." (Tamer Lectures (1938), The Philosophy of Physical Science (1939), 170)

[chiralSPO]

The following info might be outdated but on late 30s, Sir Arthur Stanley Eddington claimed that "I believe there are
15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,
914,527,116,709,366,231,025,076,185,631,031,296
protons in the universe, and the same number of electrons." (Tamer Lectures (1938), The Philosophy of Physical Science (1939), 170)

If I counted that correctly, that works out to about 1.5x10⁷⁹ :-)

[Colin2B]

If I counted that correctly, that works out to about 1.5x10⁷⁹ :-)

You missed one []

[jerrygg38]

I use a corrected Eddington number using 137.036 for the inverse fine constant whereas he used 136. The number of equivalent neutrons in the universe we see is 1.586768E78. In my latest book "The Gravitational Wave and the Dot-wave theory which I am slowly posting on my group here, the universe we see only extends to our ability to view it up to a distance defined by the time since the big bang times the speed of light. We cannot see beyond this.
   In general the universe we see is part of an ellipsoid that is revolving at 0.39C according to my calculations. According to my calculations the big bang occurred 13.7827 billion years ago.
  For anyone interested in my theory please look up my group "The Gravitational Wave and the Dot-Wave theory"

[syhprum]

The universe by definition is all there is so how can it be revolving at .39C whatever that means ?.
Does Jerrrygg38 mean that the visible universe is rotating relative to the actual infinite universe.