Older Musescore versions

Alle newer version of MuseScore can open scores produced with older Versions, with very few exceptions (I think scores produced with 0.95 or older won't open in 1.2 or later)

What OS are you using? What MuseScore version are you currently using?

It is possible to have a 1.x and a 2.x Version installed and running at the same time.
You can even have all version installed at the same time but only one 1.x Version and one 2.x Version running.

At least on Windows the scores (mscz or mscx files) will get opened by the version than got installed last, when you doubleclick them.

I have 1.3 and 2.1-RC installed, 1.3 just in case I Need to do some minor Edits to my old 1.x score and don't want to deal with the layout changes that 2.x does to them.
I also have portable apps for 1.0, 1.1, 1.2, 2.0, 2.0.1, 2.0.2. and 2.0.3, mainly for debugging purposes, to see whether a certain issue in one Version also exists in another.

You can get all those from https://musescore.org/en/download#Older-versions

You need to find out which directory you file is in if you don't know already, the use the open command in 2.0.3 (or whichever version you have) and navigate to the folder with the old score. As Jojo said, there may be some layout issues to deal with, but most are not too diffficult to fix in 2.x. Once you are done, you will want to use save as to prevent the score from thinking it is still a 1.x score. After that you can double click it from the folder and it will open with 2.x.