Hey Tilly, Here's what I've tried:

#!/usr/bin/perl -T
use strict;
use CGI qw(:standard upload);
use CGI::Carp;
use HTML::Entities;
print header(), start_html('Upload Test'), h1('Upload Test'),
  start_multipart_form(), "Enter how many files to upload: ",
  textfield("filecount"), br();
for my $i (1..param('filecount')) {
  print "File $i: ", filefield(-name => "file$i"), br();
}
print submit();
for my $file (sort grep /file/, param()) {
  print p();
  my $handle = upload($file);
open (IMAGE, ">$1") || die "Couldn't open image file"; 
    # I've tried the following variables for the filename: 
    # $1, $handle, $file, $file$i.  
    # I've also tried a generic filename (ie. test.txt)
    # I made sure that I put an absolute path name in and 
    # set the permissions to 0777.  

my $buffer;
 while (read($handle, $buffer, 10240)) {
 print IMAGE $buffer; 
} 

  unless (defined($handle)) {
    if ($file =~ /(\d+)/) {
      print h3("File request $1 did not return a handle\n");
    }
    next;
  }
  print p(), h3("Uploaded $handle"), br(), "<pre>";
  print encode_entities($_) while <$handle>;
  print "</pre>";
}
close(IMAGE);
[download]

What did I do wrong and what else can I try? All that happens is the page gets reloaded and the file field goes blank each time.