Code to upload multiple files with HTML and Flask.

Directory structure: 

dir/
├── app.py
├── files
└── templates
    └── upload.html

examples/flask/upload/app.py

import json
import os
import uuid
from flask import Flask, flash, request, redirect, url_for, render_template, send_file
from werkzeug.utils import secure_filename

# the "files" directory next to the app.py file
UPLOAD_FOLDER = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'files')
#print(UPLOAD_FOLDER)

app = Flask(__name__)
app.config['UPLOAD_FOLDER'] = UPLOAD_FOLDER
app.config['SECRET_KEY'] = 'Sick Rat'

@app.route('/', methods=['GET'])
def main_page():
    return _show_page()

@app.route('/', methods=['POST'])
def upload_file():
    if 'file' not in request.files:
        flash('No file part')
        return redirect(request.url)
    #file = request.files['file']
    app.logger.info(request.files)
    upload_files = request.files.getlist('file')
    app.logger.info(upload_files)
    # If the user does not select a file, the browser submits an
    # empty file without a filename.
    if not upload_files:
        flash('No selected file')
        return redirect(request.url)
    for file in upload_files:
        original_filename = file.filename
        extension = original_filename.rsplit('.', 1)[1].lower()
        filename = str(uuid.uuid1()) + '.' + extension
        file.save(os.path.join(app.config['UPLOAD_FOLDER'], filename))
        file_list = os.path.join(UPLOAD_FOLDER, 'files.json')
        files = _get_files()
        files[filename] = original_filename
        with open(file_list, 'w') as fh:
            json.dump(files, fh)

    flash('Upload succeeded')
    return redirect(url_for('upload_file'))


@app.route('/download/<code>', methods=['GET'])
def download(code):
    files = _get_files()
    if code in files:
        path = os.path.join(UPLOAD_FOLDER, code)
        if os.path.exists(path):
            return send_file(path)
    abort(404)

def _show_page():
    files = _get_files()
    return render_template('upload.html', files=files)

def _get_files():
    file_list = os.path.join(UPLOAD_FOLDER, 'files.json')
    if os.path.exists(file_list):
        with open(file_list) as fh:
            return json.load(fh)
    return {}

examples/flask/upload/templates/upload.html

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="utf-8">
  <meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=yes">
  <title>Upload files with Flask</title>
</head>
<body>
<h1>Upload files<h1>

    <form method=post enctype=multipart/form-data>
      <input type=file name=file multiple>
      <input type=submit value=Upload>
    </form>

{%- if files %}
  <h2>Files</h2>

  <ul>
  {%- for code, file in files.items() %}
      <li><a href="/download/{{ code }}">{{ file }}</a></li>
  {% endfor %}
  </ul>
{% endif %}

</body>
</html>

Run as: 

FLASK_DEBUG=1 flask run