In any introduction to superconductors, you can find a brief explanation of the history. The first superconductor ever discovered was Mercury, which only superconducts below 4.2 K.* As more superconductors were discovered, this temperature slowly increased, but theory predicted it would never grow beyond 30 K. And then in 1986, there was a breakthrough and high temperature superconductors were discovered!
*K is a unit of temperature relative to absolute zero. Room temperature is 290 K.
My favorite part is always the timeline, which shows the critical temperature (Tc) of superconductors vs time of discovery. It sort of looks like you could fit it with an exponential curve:
logistic function. But if it's a logistic function, that implies that we're hitting a limit on high temperature superconductivity. I'm not sure about that... we could have another breakthrough around the corner.
It sort of looks like the field of superconductivity has had large periods of stagnation. But this is not true! It appears like there's hardly any progress from 1941 to 1968, but in fact 1957 was when BCS theory was proposed as the explanation for superconductivity. That's a huge step of progress!
The lack of progress after 1986 is also illusory. A lot of progress was made in characterizing high temperature superconductors, as well as finding technological applications. And in 2008, a whole new class of iron-based superconductors was discovered, but isn't shown because it has a lower Tc than the copper-based superconductors.
I feel like this is a little lesson in predicting scientific progress. It is true that there is always progress, but if you look at any particular aspect, say the highest temperature superconductor, it does not progress the way you expect! Sometimes progress comes in jumps and spurts. And sometimes progress comes in the form of understanding the obstacles to moving forward. The question isn't necessarily, "How can we overcome these obstacles?" but instead could be, "Is it even possible to overcome these obstacles?"
A note on the graphs: I gathered the data points from various sources (mainly Wikipedia). However, I may have missed a few superconducting materials, and not all sources even agree on the dates and temperatures. Therefore, I will not vouch for the accuracy of every detail, and I don't care to list the materials, dates, or temperatures.