There are many flyfishing techniques which all aim at this:

to SUDDENLY present the lure to the fish WITHOUT showing the whirls of the line in the air.

What I just said is all so much more true when the bait is represented by a "dry" fly, the "food"-imitation which should just float ON the surface and resemble an insect fallen (or landed) from the sky (or from within the water), for instance.

Now there is nothing better before practicing than to WATCH the movements of fly casting.

"Joan Wulff's Dynamics of Fly Casting" (Miracle Productions, DVD, 90 minutes) is the way to go. It's not just me saying that. I highly recommed it for any beginner!
Joan is wife of US flyfishing legend Lee Wulff (e.g. salmon fishing flies) and is a famous fly fishing instructor, a born teacher, and in the video she show the elements of great casting: vital hand and arm movements and practice routines for learning almost all kinds of casts. They are all shown and explained, clearly and concisely, sometimes using on-stream examples and sometimes utilizing computer graphics. Lessons cover the basic cast, false casting, double hauling, roll casting, and other useful topics.

Excerpt: "A dry fly fisherman has to be aware of what we call 'drag.' Drag is that motion that makes the fly look as if it is a water-skier instead of a free-floating insect, and it's caused by the currents that work on the line and leader. So we need affect that by doing what we call a 'Reach Cast,' which will put the arm and the upper part of the rod upstream of the fly, so that the fly drifts down first."

Reviews:

"This video is absolutely wonderful for beginner fly fisher/casters...."
"This DVD is so beautiful..."
"A teacher's heart..."
"I am a complete beginner, but with this video as my guide, I am already casting fairly well after only four practice sessions..."
"Excellent step by step from the basics to advanced..."
"Joan Wulff is the Best!" (from a fly fishing guide!)
"The production qualities of this video are excellent: the audio is as clear as a bell, and the scenery is beautiful"
"Buy it!"

Buy "Joan Wulff's Dynamics of Fly Casting" ($24.95) from Amazon

fly casting techniques
More theroy to read. Fly casting make fly-fishing so beautiful to look at (do you remember the movie a river runs through it?) but it can be a higly sophisticated sport. Those who master are like musicians, it seems easy to do because they make it look so natural...

Spey casting Techniques
You can use a special, longer, two-hands rod that you move with special techniques....Lots of advantages with this technique!

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Wanna give a look at a rod ACTION?

....or do you need any tips on fly-fishing reels?