I have a 2005 Dodge Neon. The heater stopped working. The blower still works so I am pretty sure the fuse is ok, at least I can’t find a separate fuse for the heater. All I see is the blower for the AC/Heater. I replaced the Thermostat and watched it working by letting the car heat up and watched the engine heat drop after a bit. Still no warm air out of the heater. I tried flipping between Cold air and hot air and can tell a difference in the sound of the air blowing that SEEMS to indicate that the flap in the Dash is working to go from cold to hot air. I am sort of out of ideas unless the heater itself is hosed. Any thoughts?
I will have to check that right after I figure out where the heater hoses are. 
I have reached the end of my limited knowledge on this. What if they are and are not? What does that tell me?
If only one heater hose is hot, your heater core is plugged
@Glupii, there should be two 5/8" rubber hoses going into the firewall right about where the middle of the dashboard is. These are the hoses for the heater core. I hope it is just clogged, because the entire dash would need to come out to replace it.
If you just replaced the thermostat, then look right below the water outlet (the black plastic thing you pulled off to get to the thermostat). Under it is a little stub pipe that comes out from the engine. You’ll need a flashlight. Attached to that stub is a coolant hose. Follow it. It runs underneath of the intake manifold, out the right side and around the corner of the valve cover. (Just in case - The valve cover is the black thing on top of the engine with oil cap on it, though the oil cap is at the opposite end). There’s also a second hose running under the intake though harder to see. But you’ll find both of them running out from under the intake, around the corner of the valve cover, and then between the valve cover and the battery. Those are the heater hoses. Follow where they go and you’ll see where they hit the firewall. Behind the firewall where they connect is your heater core.
You can feel them for temp there next to the battery where they are easy to get a hand on. After the car is all warmed up they should both feel hot.