Can you run Romex exposed outside?

Can you run Romex exposed outside?
No. Romex is not allowed to run exposed in any situation. In a dry, exposed area, you can use MC cable and metal boxes. In a damp area as your patio cover could be, you could use non-metallic seal-tite or the trade name is Carflex or EMT.
What type of wire do you use to run outside?
Type UF cable is the most commonly used nonmetallic cable for residential outdoor wiring runs. UF cable can be direct-buried (without conduit) with a minimum of 24 inches of earth cover.
Can you run Romex and run it in a conduit?
When using indoors, you have the freedom to choose. When using any non metallic wire outdoor, including Romex, it is a good idea to run it through a conduit. Yes, they can be used outdoors, it is better to be safe than sorry.
Is it OK to use Romex in outdoor lighting?
It's rather common to protect or shield Romex with PVC in locations where (wet or not) it might be harmed by objects or people, but if you're going to be running high voltage (120 V+) outdoor lighting or receptacles it's a much better idea to just use UF, or PVC conduit and pull the THHN conductors through it as you need.
Can you run nm-b cable outside of a conduit?
Since @Tim's post didn't make this clear: under the current code, you are not allowed to run NM-B wire outside, even through a conduit. That is, you are no longer allowed to put NM-B cable in watertight conduit outside of the house in wet locations.
What kind of UF cable is Romex for?
Romex is a brand name for non-metallic paper bonded cable (NM-B, as printed on the jacket) with a PVC jacket. This is why type UF is suitable for burying and wet locations (or dry) where NM-B cable is suitable only for very specific dry locations.