<?php
$router->rewrite('/some/arbitrary/url/{slug}', function($slug) {
  // you can do anything here
  update_option('option_name', $slug);
  // if you return false, the request is over
  //return false;
  // if you return a string, WordPress will try to load a template by that name
  return 'my-custom-template';
});

// you can also use Router::rewrite for more traditional rewrites to index.php
$router->rewrite('/some/other/url/{slug}', 'index.php?slug=$matches[1]');

// and if you want to add to the REST API, that's easy too:
$router->get('/my/custom/endpoint', function() {
  // you can do anything here
  // and whatever you return will be transformed into JSON
  return [ 'foo' => 'bar' ];
});