It all depends on your objectives as a researcher. If you care a lot about getting the right answer then you may not be willing to accept any bias, as in OLS and most related models. However, there are many situations in which you might prefer to get as close as you can to the right answer, where closeness is the priority and unbiasedness is not. In this case you can often do better by minimizing the mean square error, which is the sum of the variance and the squared bias, than you could by searching for the minimum variance unbiased estimator, which is what most simple regression models do.
The question is very broad, so maybe this discussion can help narrow down what you're looking for.