How to submit an HTML form without redirection

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);
}

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>