Using Low Search for ExpressionEngine with Ajax. Change any of the three variables on top to suit your needs. Tested with jQuery 1.9.0.

low-search-ajax.js

(function($){
	$(function(){
		var $form   = $('#search'), // Search form
		    $target = $('#results'), // Results container
		    rp      = 'search/ajax-results'; // Template for results only

		// Function to execute on success
		var success = function(data, status, xhr) {
			$target.html(data);
		};

		// Hijack the form on submit
		$form.submit(function(){

			// Tell target we're loading
			$target.html('<p>Loading...</p>');

			// Add custom result page to data
			var data = $form.serialize()
			         + '&result_page=' + rp;

			// Perform the ajax call
			$.ajax({
				type: 'POST',
				url: $form.attr('action'),
				data: data,
				success: success,
				dataType: 'html'
			});

			// Don't submit the form afterwards
			return false;
		});
	});
})(jQuery);

Use secure="no" in the Form tag to prevent "not authorized to perform this action" error message.

Also, make sure your Redirection Method in the Output and Debugging settings is set to "Redirect", not "Refresh".

First result in google. Thank you Low! 👍

Is it possible to get the encoded query so we can change the URL via pushstate?

"Is it possible to get the encoded query so we can change the URL via pushstate?"
+1

What should the search results tag look like within the search/ajax-results template? eg:
{exp:low_search:results query="{last_segment}" channel="{segment_1}" orderby="title" sort="asc" }

Hi Low, this is great thank you. Any pointers to integrate pagination?

@low Does this still work in the EE4 version?

@low I'd also be interested to know if this works in the EE4? @mindpixel-labs did you get it to work?

This approach still works today, but you might want to tweak the jQuery syntax (or use some other JS lib).