Solar System -- after 500 years of research, multi-body problem still not solved, many features still amaze and baffle, scientists have no idea how common the system might be
Chemistry -- in spite of having a great model with lots of reproducible experiments, issues remain: http://en.wikipedia.org/wiki/Unsolved_problems_in_chemistry
Biology -- where to start?
Evolution -- big vague term. Hard to critique this one. Do you mean natural selection inside a species? Speciation?
Medicine -- wonderful ability to find and fix many diseases. Lots of things work although we have no idea why. Lots of things that we think should work don't. Lots of medicine is so much of a mystery that only statistical analysis can give us rough correlations to begin to form theories. The reason it is so advanced? Billions of test subjects (as opposed to one climate for Earth)
And let's not even get into physics, quantum or otherwise. Or cosmology. Or the problems with the use of Cellular Automata as a modeling tool. I could go on.
Be careful with grand sweeping statements. Ignorance is a fine position and should be the default. Science is a humble questioning of the cosmos, always willing to correct itself and always looking for reproducibility. Be careful about putting so much trust in scientists that you forget what science is about.
Ignorant people have plenty of tools at their disposal to reason and understand the quality of the underlying science. The science itself may be complicated beyond belief, but the underlying meta-data -- how science is done -- is not.