It sounds like you need a meta_query
.
$query = new WP_Query (
array(
'paged' => $paged,
'post_type' => 'event',
'posts_per_page' => 4,
'meta_query' => array(
array(
'key' => 'date_key_name',
'value' => date('Y-m-d'),
'compare' => '>=',
)
)
)
);
Be aware that your dates need to be in some format that a machine can parse correctly.
Reference
##Reference http://codex.wordpress.org/Class_Reference/WP_Query#Custom_Field_Parameters