Introduction

Starting with vB4, templates no longer get output using eval:
eval('$mytemplate = "' . fetch_template('mytemplate') . '";');is outdated.
What's more: Variables and arrays from plugins that are executed on a page no longer can automatically be accessed in the templates of that page. They need to be registered first.
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Basic functionality to render templates and register all variables/arrays you want to use inside

/* Some Code, setting variables, (multidimensional) array */
$my_var = "abc";
$my_array = array(
'key1' => 'value1',
'key2' => array('
'key21' => 'value21',
'key22' => 'value22'
')
);

/* render template and register variables */
$templater = vB_Template::create('mytemplate');
$templater->register('my_var', $my_var);
$templater->register('my_array', $my_array);
$templater->render();

The first line provides the template that is to be rendered, using the new vB_Template class (vB_Template::create). The method gets passed the name of the template as an argument.
The following two lines register a variable and an array that we want to use in our template. Arguments passed are 1. the name you want to use to access the variable, and 2. the variable from the code you want to register. You can register as many variables/arrays as you want. Just remember you have to register every variable and array that you want to use in your custom template in this way. If you don't register them, they will not be available.
The fourth line renders the template ($templater->render()). In the template you know will be able to use the registered variables/arrays in this way:
{vb:raw my_var}
{vb:raw my_array.key1}
{vb:raw my_array.key2.key21}Note the last one: multidimensional arrays are perfectly possible.
.
.
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Now, with the result of the rendering we can do several things:
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Output template directly - custom pages

$templater = vB_Template::create('mytemplate');
$templater->register_page_templates();
$templater->register('my_var', $my_var);
$templater->register('my_array', $my_array);
print_output($templater->render());This immediatly outputs the template. Use this if you have created your own page, for example.
Note the second line, which is special for this type of use:
$templater->register_page_templates();This auto-registers the page level templates header, footer and headinclude that you will use in the template of your custom page.
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Use a template hook

$templater = vB_Template::create('mytemplate');
$templater->register('my_var', $my_var);
$templater->register('my_array', $my_array);
$template_hook[forumhome_wgo_pos2] .= $templater->render();The template will be shown using the choosen template hook (for example: $template_hook[forumhome_wgo_pos2]). See the dot before the = in the last line? The hook may be used by other modifications, too, so we don't want to overwrite it, but rather append our code to it, conserving everything that might already be there.
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Save into a variable for later use in custom template

$templater = vB_Template::create('mytemplate');
$templater->register('my_var', $my_var);
$templater->register('my_array', $my_array);
$mytemplate_rendered = $templater->render();Now we have saved the rendered template into a variable. This variable in turn we can later on register in another template, if we want:
$templater = vB_Template::create('my_other_template');
$templater->register('my_template_rendered', $my_template_rendered);
print_output($templater->render());Again, inside my_other_template we now could call
{vb:raw my_template_rendered}If you're running the first template call inside a loop, you may want to use .= instead of = in the last line, so that the results of every loop get added instead of overwriting the existing. But that depends, of course.
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Save into an array and preregister to use in an existing/stock template

$templater = vB_Template::create('mytemplate');
$templater->register('my_var', $my_var);
$templater->register('my_array', $my_array);
$templatevalues['my_insertvar'] = $templater->render();
vB_Template::preRegister('FORUMHOME', $templatevalues);

This is another, more flexible method to save the rendered template into a variable for future use in an already existing template. In this example, we want to show our own rendered template on forumhome.
Problem is: We have no direct way to register variables for already existing templates like FORUMHOME. It's created and rendered in the files, and we don't want to mess there.
To help with this, a new method was created for vB_Template class, called preRegister. Using this, we can pass our data to FORUMHOME before it is rendered. Note that the data needs to be saved into an array ($templatevalues['my_insertvar']), a simple variable will throw an error. In the last line the array is preregistered; you need to pass as arguments 1. the name of the existing template and 2. the array that contains the data. Again, this can be done for as many arrays as needed.
Of course, the preRegister functionality can be used for any kind of variables or arrays, no matter whether you have saved a rendered template (like in our example) into it or it contains just a simple boolean true/false statement. To access the data inside the template it was preregistered for use:
{vb:raw my_insertvar}Note: it is not {vb:raw templatevalues.my_insertvar}!

Essentially the same as what I put for preRegister would be the following two lines. They could replace the last two lines in the above php codebox:
$my_insertvar = $templater->render();
vB_Template::preRegister('FORUMHOME',array('my_insertvar ' => $my_insertvar));
Of course you could add further pairs to that array if you need to preregister more than one variable.
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Bonus track: ...whatever you do, cache your templates!

Now you know how to get your templates on screen - once you succeeded in doing that, make sure to do it in a fast and ressource saving manner: make use of vB's template cache. To see whether your templates are cached or not, activate debug mode by adding $config['Misc']['debug'] = true;to your config.php (don't ever use that on your live site!). Among the debug info is a list of all templates called, and non-cached templates will show up in red.

To cache your templates, add a plugin at hook cache_templates with the following code:

// for a single template
$cache[] = 'mytemplate';

// for more than one templates in one step
$cache = array_merge((array)$cache,array(
'mytemplate',
'myothertemplate',
'mythirdtemplate'
));.
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Hope this helps!
-cel

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Addendum - There are now two blog posts on vb.com related to this topic:
http://www.vbulletin.com/forum/entry.php?2387-Pushing-your-variables-to-vBulletin-4-templates
http://www.vbulletin.com/forum/entry.php?2388-Using-custom-templates-within-your-own-vBulletin-4-based-files