Why threre is no way to download file using ajax request?

This can be done using the new HTML5 feature called Blob. There is a library FileSaver.js that can be utilized as a wrapper on top of that feature.

That's the same question I'd asked myself two days ago. There was a project with client written using ExtJS and server side realisation was on ASP.Net. I have to translate server side to Java. There was a function to download an XML file, that server generates after Ajax request from the client. We all know, that it's impossible to download file after Ajax request, just to store it in memory. But ... in the original application browser shows usual dialog with options open, save and cancel downloading. ASP.Net somehow changed the standard behaviour... It takes me two day to to prove again - there is no way to download file by request usual way ... the only exception is ASP.Net... Here is ASP.Net code

public static void WriteFileToResponse(byte[] fileData, string fileName)
    {
        var response = HttpContext.Current.Response;

        var returnFilename = Path.GetFileName(fileName);
        var headerValue = String.Format("attachment; filename={0}", 
            HttpUtility.UrlPathEncode(
                String.IsNullOrEmpty(returnFilename) 
                    ? "attachment" : returnFilename));
        response.AddHeader("content-disposition", headerValue);
        response.ContentType = "application/octet-stream";
        response.AddHeader("Pragma", "public");

        var utf8 = Encoding.UTF8;
        response.Charset = utf8.HeaderName;
        response.ContentEncoding = utf8;
        response.Flush();
        response.BinaryWrite(fileData);
        response.Flush();
        response.Close();
    }

This method was called from WebMethod, that, in turn, was called from ExtJS.Ajax.request. That's the magic. What's to me, I've ended with servlet and hidden iframe...

It's not about AJAX. You can download a file with AJAX, of course. However the file will be kept in memory, i.e. you cannot save file to disk. This is because JavaScript cannot interact with disk. That would be a serious security issue and it is blocked in all major browsers.