Analysis of accuracy data? I have accuracy data, where every subject gives a correct, i.e. 1 or wrong, i.e. 0 answer. However every subject performs 190 trials in each session, with 10 sessions in total. 

To analyze the data I use R, and group the results by session, i.e. more simply put every subject responds 190 times in each session, however after grouping, it gets only one value for each session (I use the group_by function of package dplyr), based on the average of all its trials in that session (could be 1.00 if all the answers are correct or 0.95, 0.85 etc. if the subject has mistakes).

My reasoning is that in this way the data is no longer categorical, but quantitative, therefore I was hoping to use repeated-measures ANOVA for the anlysis. However the normality tests that I did on the data showed to be signifcant and a graphical representation of the data also shows that is very strongly skewed to the right (wich is completly logical of course, since there are much more correct answers than wrong).

My question is: What can I use to analyze this data and what is used in the scientific community in general in those situations, since it is not an uncommon problem, accuracy being often measured? I checked some articles, but there for accuracy researchers simply report percentage differences, and state that they are (or aren't) significant.  I know that I could use Friedman's ANOVA, but it isn't that powerfull so I'm looking for another way. Also researchers often report using ANOVA for accuracy scores, which is leaving me absolutely stunned...
Maybe there are some errors in my reasoning, so all comments are welcome!!

Thank you for the answers!!!
ANOVA and linear regression are inappropriate when your outcome is a percentage of correct trials, especially so if numbers are near the boundaries 0% or 100%. 

Mixed effects logistic regression may be appropriate. 

More detailed answers would require more details about your experiment.