In a Dutch oven, melt the fat over medium-low heat, then cook the onions until translucent, stirring, about 10 minutes.
Add the beef and cook until no longer pink on the outside, about 10 minutes.
Add the garlic and caraway, season with salt, stir.
Add the paprika, beer, and water, and stir to blend everything.
Bring to a gentle boil, then reduce the heat to low, cover, and simmer for 2 hours.
Add the tomato and peppers and taste the stew to see if it needs any more salt (usually, at least a teaspoon).
Increase the heat to medium and cook until the peppers are soft, about 1 hour.
Add the potatoes and cook until everything is tender, about 1 hour more, stirring occasionally.
The goulash will be thicker, similar to a stew.
Taste and adjust seasonings.
Serve, if desired, with Hungarian Dumplings (recipe below)