The Big Bang
To understand the shape of the Universe, we have to start with the Big Bang. Some time about 13.7 billion years ago, a cosmological event occurred that created very high levels of energy. The heat energy was so high that it took between 300,000 to 500,000 years for it to cool down and create atoms. This was the Big Bang. During that time period there was a brief period of growth, called the inflationary period, which took the Universe from the size of less than an atom to about 5 times the size of a galaxy. (The Milky Way Galaxy is about 5.90 x 1017 miles across). The inflationary period lasted only a brief moment from 10-36 seconds after the big bang to 10-32 seconds.
This image taken by the COBE satellite shows the early universe about 400,000 years after the Big Bang.
This image shows the sequence of the Big Bang.
See Also: How Old is the Universe?
The Size of the Universe
If you ask the question, how big is the Universe, you look at its size since the Big Bang. The size of the Universe is one of the most interesting questions astronomers and cosmologists ask. That is because the size keeps increasing. Even now, scientists have discovered that the size is changing faster than before. The study of the changing size of the universe was first examined by the American astronomer, Edwin Hubble. He noticed that by studying galaxies, he observed that they were moving away from our galaxy; that their light was red-shifted. He concluded that the Universe was expanding. Eventually it was learned that the rate at which the Universe is expanding has changed over space and time.
Currently, the distance from Earth to the edge of the observable Universe is about 14 billion parsecs (46.5 billion light-years) in any direction. This makes the observable universe look like a sphere with a diameter of about 28 billion parsecs (93 billion light-years).
What does the Universe look like?
The dimensional appearance of the Universe depends on the density. In order to establish the difference between the shapes of the Universe, you have to look at the critical density. This is the amount of mas which will determine if the Universe will close in on itself, like a sphere, or not enough mass to stop the expansion and the Universe will continue to expand.The critical density tells you how much mass you need to keep the Universe flat.
The density of the Universe is a factor that determines its geometry. If the density of the Universe exceeds the critical density, approximately 10-30 grams per cubic centimeter, then the geometry of space is closed, and it has a positive curve like the surface of a sphere. If the density of the Universe is less than the critical density, then the geometry of space is open, it has a negative curve like the surface of a saddle. Finally, if the density of the Universe exactly equals the critical density, then the geometry of the Universe is flat like a sheet of paper. This tells us that there is a direct link between the geometry of the Universe and its fate.
Most astronomers are still trying to answer the question about the critical density of the Universe. That said, many believe that the density of the Universe is near the critical density and its shape is close to being flat.
See Also: Nothing But the Facts About Cosmology
The Big Bang: https://www.mhest.com/spotlight/darkmatter/articles/AccessScience_Cosmology.pdf
Image #1: https://www.nasa.gov/topics/universe/features/cobe_20th.html
Image #2: https://map.gsfc.nasa.gov/universe/bb_tests_cmb.html
The Size of the Universe: https://www.physicsforums.com/showthread.php?t=440065
What does the Universe Look like: Image https://www.physics.hku.hk/~nature/CD/regular_e/lectures/chap19.html