Before you do anything to your current install installation of PrestaShop, you should check that your server configuration matches or exceeds the minimum requirements for the latest version of PrestaShop, which are indicated on this page: http://www.prestashop.com/en/system-requirements.
For instance, make sure that your web host provides does provide you with the required versions of PHP and MySQL. If not, ask him your host to update the server configuration. If you are unsure which PHP and MySQL versions you are using, ask your web host.
If you web host won't update your server configuration, then it is time for you to find a better host. Do NOT perform an upgrade unless your server at least meets the system requirements.
The minimum requirements for the latest version of PrestaShop are available on the official PrestaShop download page.
If upgrading from a version below 1.4.x
Older versions of PrestaShop might not make it as easy to upgrade to the latest version: the code base is quite different, many files have been moved around, created or deleted, and many have been created, and others have been deleted. Therefore, the risk is high, when skipping from a very old version to the latest one, to see errors and mistakes aplenty.
You are also limited to the manual update process: the AutoUpgrade module can only work with version 1.4 and up.
You should therefore be even more careful; the older your version, the more you should pay attention to details: backups, custom files, edited theme, etc.