A hammer or a quick slap will kill an insect, but how do you control lots of them in your garden? My father paid me a quarter to hand-pick potato bugs in our garden but there is currently a shortage of impecunious eight-year-olds.
Organic insecticides vary in their effectiveness but they can be useful if used early, before a big outbreak. Aphids can be controlled with a strong stream of water. Give a kid a Super Soaker and have them blast away!
I found an excellent list of organic alternatives for your garden. Download it and study the details.