If you spend a lot of effort in having a vegetable or flower garden, seeing aphids on prized plants sends most people out to their local Bunnings for conventional or organic insecticides. Using insecticides will kill prey insects like aphids that eat your plants but also predator insects such as ladybugs that feed on prey insects. It may be better just to leave the plants alone and let the predator insects do their job.
Homesteading magazine Mother Earth News' weblog makes the case that using insecticide is a bad idea in the long run as prey insects multiply quickly enough that after a few generations (for insects this can be just a few years or even months) they have built up an immunity to the insecticides. Plus, you've killed off beneficial predator insects as they have a lower reproductive rate than prey insects and are more susceptible to being wiped out by insecticides.
Another general rule is that prey reproduce better than predators. If they didn't they would go extinct. Ladybugs make it from egg to larval (predatory) stage in about 5 days from as few as 3 eggs per "nest". It takes a ladybug up to 6 weeks to reach adult (egg laying) stage. Conversely, an aphid hatches from a nest of 50-100 eggs. The aphids that hatch only take about a week to reach adulthood (and thus egg laying potential). And that disregards the fact that if the aphid female cannot find a mate it can simply clone itself (in a process known as parthogenesis) and give birth to a live juvenile copy! So I ask you, if you nuke both insect groups, which will come back quicker?
Also if you use an organic spray to kill or get rid of prey insects you are eliminating the food source for predator insects who are then much less likely to remain in your area.
It can be difficult to wait a few days for predator insects to enact while watching aphids eat your plants, but it may be the best (and least expensive) strategy to save the most plants in the long run.
The Case for Doing Nothing about Pests [Mother Earth News]