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I just trained up for a Minmatar Slasher to use its extreme speed and maneuverability in my courier missions. After fitting it properly, I learned that I can push 1,000m/s velocity, and outrun just about anyone. I realized that if I can keep a great enough distance, I should be able to gun ships down with missiles without having to go in range of their fire. What kind of missiles do you recommend, and what's about the max range I can get to safely shoot down enemy ships with?

Well, first of all: Train Missile Bombardement and Missile Projection both will up your range (by increasing flight time or velocity of the missile by 10% per level each).

Other than that, you can only upgrade to other kinds of missiles - each and every light missile has the same range (flight time * velocity ~ 19 km) - not couting advanced missiles, the heavy payload ones have reduced range.

Maybe you can fit Heavy Missile Launchers on the Slasher, HM have a standard range of ~37 km.
 

 Slasher won't fit heavies, it has very low fitting resources even for a frigate. Light missiles (standard launcher) are probably the best option, otherwise it's rockets (really short range, but higher dps). Light missiles can hit out to 42km with max skills (18km with no skills); training the skills janek suggested will do the trick. Rockets have an absolute max of about 10km.

 Of course bear in mind the slasher only has 1 missile hardpoint anyway... If you want a fast missile frigate, you may be better looking at the caldari condor (2 missile hardpoints, but not as fast as the slasher) or the minmatar breacher (not that fast for a frigate, but more missiles again).

 Also, go for a microwarpdrive if you want speed, ideally with an overdrive injector in the low slot (or a nanofiber internal structure - not as much speed increase, but faster warping and turning). With those, a slasher can get well over 4km/s!

Considering this velocities you are speaking about Azual, I'm really happy that EVE doesn't have ramming damage... (Or does it?)

You also shouldn't forget the Minmatar Rifter, also with 2 missile hardpoints, unboosted 310 m/s. 

 No ramming damage, no, but ramming can be used to 'bump' ships suprising distances in a fast enough ship. It can be quite useful to say... knock someone outside dock range of a station/jump range of a gate.

Wow, so with full skills you could keep at 40+km range and snipe someone down while keeping enough speed to keep them from getting a shot off you. And ramming sounds like it would be a lot of fun too. I'd love to rocket into a belt and push people around just to watch their ships spin out of control. This would be especially fun for freaking out AFK miners... or giving Artisan a taste of his own medicine.