I know that the distribution of sample mean of a number of iid random variables tends to a Gaussian distribution if the sample size is at least 30, where the mean value of the resultant distribution turns out to be the same as the population mean of the iid random variables. I need to know whether any such results exist about sample median of a number of iid random variables? Can anything be said about the distribution of the sample median of $n$ iid random variables where $n \ge 30$?
The sample median, if you are lucky (if your sample size is odd), can be an order statistic. So given that you know the distribution of your original sample you can order it as the Wikipedia article describes and then the middle observation has a specific cumulative distribution function and potentially a probability density function. Practically even if your sample size is even you can calculate the CDF of the middle value (call it $m$) and the $m+1$ and average them out for practical purposes it should be fine.