You have decided to use a boxplot to summarize each of the 20 distributions.
A boxplot is a 5-number summary: it shows the minimum, lower quartile, median, upper quartile, and maximum. It has advantages (it's simple and well-known) as well as disadvantages (How should we do boxplots with small samples?).
There are other ways to visualize a sample from a distribution and, depending on your actual data, one of the alternatives may work better than a boxplot: If there are a few observations from each distribution you can show the raw data in a Wilkinson dot plot (aka a stacked dot plot). If the samples are larger (at least ~30 points per distribution), you can make a histogram (with 10-12 bars since there are 20 histograms to show) or a kernel density (smooth histogram).
There is also the question how to put together the individual representations of 20 distributions. A small multiples graphics, with several panels arranged in a grid, can be very effective. In your case there will be 20 panels, one for each distribution, so it's necessary to pare down the details in each panel. However the juxtaposition with common x- and y-axes makes comparisons between distributions very intuitive.
Here is an illustration. I draw 50 observations from 20 distributions: 16 distributions are Normal(0,1) and 4 distributions are Normal(1,2). I estimate and plot the density of each sample and use color to highlight the "unusual" distributions. You can use color to indicate different experimental conditions or the levels of a categorical variable, if relevant.
This example is inspired by David MacKay's Information Theory, Inference, and Learning Algorithms. This book is full of amazing graphics, all without color. Chapter 21 has several small multiples plots which represent different combinations of the mean $\mu$ and variance $\sigma^2$ of the Normal distribution, in a mixture of two Normals.
 D. J. MacKay. Information Theory, Inference, and Learning Algorithms (2003). Available free online.