...Take the progression Am
F C G
That list of chords doesn't determine a key.
Those chords as a bunch of tones could match the key signatures of A minor or C major... or you could even say it's in F where the G chord is a secondary chord, etc.
etc.
The real point of putting Roman numerals on the chords it to say what key the chord are in!
IMO it's more important to say what the key is than exactly which chords are used, because key structure is paramount in understanding compositional form.
Put a key label with the Roman numerals...
Am:
i VI III VII or C: vi IV I V
However, you need to consider one other important point.
The traditional way to define a key is with a dominant chord , and the clearest way to do it is a progression from a dominant to tonic chord.
So, you would want either Am: V I or C: V I somewhere in the music to make a clear case for a key actually being established.
If you don't establish a key with a dominant to tonic the music can be described as non-functional harmony .
That sounds bad, but it isn't.
It's simply a harmonic description.
Lots of pop & rock music is non-functional.
But, when the harmony is non-functional - lacking a clearly defined key - it sort of become irrelevant which Roman numeral labels you use.
music.stackexchange.com/users/23919/Michael Curtis
My rule of thumb is: is the tonal centre of the song Am ?
Stick to i VI III VII .
Does the song start in the relative major key for instance ( C in this case), then feature this passage?
Then stick to vi IV
I V .
See Hallelujah by Leonard Cohen , the verse is in the major key (starts with the I/vi vamp, then explicitly says it goes like this, the 4th the 5th ...), and goes vi/IV in the chorus.
music.stackexchange.com/users/35439/moonwave99
Depends if you choose to relate it to the key of C major or A minor.
If you choose A minor - and if A is established as the tonic you probably should - there's then the additional question of whether to call C III or &flat;III.
I know which I prefer - after all, if you call C III, what would you call C&sharp;?
(It COULD happen!)
But some people disagree, and it's pointless to brush aside an established practice as merely 'wrong'.
music.stackexchange.com/users/17003/Laurence Payne
Am F C G - in key Am will make Am i .
F is VI , C is III , G is VII .
In key C it will make Am vi , F is IV , C is
I , G is V .
Usually the telling chord in Am is V - E.
So if the piece has some E chords, it's more likely to be in key Am.
If not, it'll more likely be key C.
The last chord is also a good clue to the key.
That's what will be the decider as to how you designate the Roman numerals.
It's maybe not a good idea to call it a minor chord progression -
there's ony one in there!
Chord progression in a minor key make it clear that it's not in key C!
music.stackexchange.com/users/2808/Tim
-> The axis of awesome 4 chords songs
Your example starts with Am: so this one i VI III VII and it is modal.
Compare it with: https://en.wikipedia.org/wiki/I%E2%80%93V%E2%80%93vi%E2%80%93IV_progression
music.stackexchange.com/users/56219/Albrecht Hügli
I'm not sure what the context of your situation is, but if you can make those chords more than a triad (such as adding dominant/minor 7ths, etc.), it could be possible to determine the key.
For example, if: Am -> Am7 F -> Fmaj7 G -> G7 C -> Cmaj7 , then it would definitely be in C.
The pattern of this in any given major key is: I -> maj7 II -> m7 III -> m7 IV -> maj7 V -> 7 VI -> m7 VII -> (half diminished)
The dominant is a great indicator because it's easier to recognize (in my opinion) and is the only one of its kind.
Albrecht Hügli is partially right in his answer, though.
This looks like a form of a standard 4 chord progression.
It shows up in some form in a large portion of modern songs.
In that wikipedia page: https://en.wikipedia.org/wiki/I%E2%80%93V%E2%80%93vi%E2%80%93IV_progression
This chord progression exactly matches the example given for the variation termed the "sensitive female chord progression" with the numeral expression "vi–IV–I–V".
music.stackexchange.com/users/66276/tlake29
Quick question,  Take the progression Am F C G   Would the correct way of writing this in Roman Numerals be   i VI III VII   Or  vi IV I V  Thanks
Roman Numerals for minor chord progression E.g. Am F C G
music.stackexchange.com
<theory><harmony>
music.stackexchange.com/questions/95490
music.stackexchange.com/users/66518/blooom
Am F C G in key Am in Roman Numerals is i VI III VII and In key C is vi IV I V In key Am, the telling chord in Am is V - E. 
In key Am, the telling chord in Am is V - E. If the piece has some E chords, it's more likely to be in key Am. If not, it'll more likely be key C. The last chord is also a good clue to the key.