Currently rivers have a start and end point. The start is shown as a spring, the end on the cloth map looks like the mouth of a river, but only on the cloth map and not actually attached to any body of water:

I think rivers should (at least usually) end either at the coast or at a lake or large body of water.

So many real cities and people have used rivers as ports and a route out to sea. Wikipedia has this to say:

An estuary is a partly enclosed coastal body of brackish water with one or more rivers or streams flowing into it, and with a free connection to the open sea...Of the thirty-two largest cities in the world, twenty-two are located on estuaries[13]
[13] ^ Ross, D. A. (1995). Introduction to Oceanography. New York: Harper Collins College Publishers. ISBN 978-0-673-46938-0.

Simply put I think more rivers should be estuaries. Why?

Realism is only part of it, but still important. Since 2/3s of big cities are on estuaries not having estuaries doesn't fit with the real experiences of rivers for 2/3 of city-dwelling folk.
Secondly. Game design. Sorcerer King has very little visual 'fluff'. Almost everything shown has a real impact on the game, the general "if you can see it you can use it" design philosophy is key and elegant but overlooked in the case of rivers. Which seem at least to have no use, no way of interacting with them and no effect on the game.

Thirdly. Fun. I think it'd be fun to mess with rivers just like messing with mountains.

Suggestions:
1. Have rivers touch the coast, with a foaming-water river-mouth tile.
2. Have spells to create rivers (possibly even just as the visual part of the 'fertility' spells.)
3. Have spells and events to destroy, harvest or otherwise dry-up rivers (turning them pale grey, with a brown/yellow or black close-up)... possibly lowering the fertility of tiles touching the river.
4. Have rivers impact movement again, not much but at least as much as forests do. Preferably with a tooltip. If pathfinding can handle forest-movement costs then it should handle rivers with the same movement cost. (Even if it couldn't handle 'rivers end movement' from early Elemental games).
5. Have bridges where roads cross rivers, with normal movement costs.
6. Give rivers names.
(7. Give mountains and seas names too - so nearby cities can use them in their name)

Basically I'd like to be able to have capital and important cities like London, Cambridge (literally Bridge over the river Cam) in-game. Arising naturally through fertility and rivers. If you saw a river, you'd know it leads to the coast and that somewhere along its length there would be fertile/more fertile terrain. It would also be nice to be able to sacrifice rivers in your territory for massive gains even if it causes the city nearby to wither and die.