How to submit an HTML form without redirection

You can achieve that by redirecting the form's action to an invisible <iframe>. It doesn't require any JavaScript or any other type of scripts.

<iframe name="dummyframe" id="dummyframe" style="display: none;"></iframe>

<form action="submitscript.php" target="dummyframe">
    <!-- Form body here -->
</form>

In order to achieve what you want, you need to use jQuery Ajax as below:

$('#myForm').submit(function(e){
    e.preventDefault();
    $.ajax({
        url: '/Car/Edit/17/',
        type: 'post',
        data:$('#myForm').serialize(),
        success:function(){
            // Whatever you want to do after the form is successfully submitted
        }
    });
});

Also try this one:

function SubForm(e){
    e.preventDefault();
    var url = $(this).closest('form').attr('action'),
    data = $(this).closest('form').serialize();
    $.ajax({
        url: url,
        type: 'post',
        data: data,
        success: function(){
           // Whatever you want to do after the form is successfully submitted
       }
   });
}

Final solution

This worked flawlessly. I call this function from Html.ActionLink(...)

function SubForm (){
    $.ajax({
        url: '/Person/Edit/@Model.Id/',
        type: 'post',
        data: $('#myForm').serialize(),
        success: function(){
            alert("worked");
        }
    });
}

Since all current answers use jQuery or tricks with iframe, figured there is no harm to add method with just plain JavaScript:

function formSubmit(event) {
  var url = "/post/url/here";
  var request = new XMLHttpRequest();
  request.open('POST', url, true);
  request.onload = function() { // request successful
  // we can use server response to our request now
    console.log(request.responseText);
  };

  request.onerror = function() {
    // request failed
  };

  request.send(new FormData(event.target)); // create FormData from form that triggered event
  event.preventDefault();
}

// and you can attach form submit event like this for example
function attachFormSubmitEvent(formId){
  document.getElementById(formId).addEventListener("submit", formSubmit);
}

Place a hidden iFrame at the bottom of your page and target it in your form:

<iframe name="hiddenFrame" width="0" height="0" border="0" style="display: none;"></iframe>

<form action="/Car/Edit/17" id="myForm" method="post" name="myForm" target="hiddenFrame"> ... </form>

Quick and easy. Keep in mind that while the target attribute is still widely supported (and supported in HTML5), it was deprecated in HTML 4.01.

So you really should be using Ajax to future-proof.