In addition to the pure javascript answers above, You can use jQuery text method as following:

$('#myspan').text('newtext');

If you need to extend the answer to get/change html content of a span or div elements, you can do this:

$('#mydiv').html('<strong>new text</strong>');

References:

.text(): http://api.jquery.com/text/

.html(): http://api.jquery.com/html/

You may also use the querySelector() method, assuming the 'myspan' id is unique as the method returns the first element with the specified selector:

document.querySelector('#myspan').textContent = 'newtext';

developer.mozilla

document.getElementById("myspan").textContent="newtext";

this will select dom-node with id myspan and change it text content to new text

You can do

 document.querySelector("[Span]").textContent = "content_to_display"; 

Can't be used with HTML code insertion, something like:
var a = "get the file <a href='url'>the link</a>"
var b = "get the file <a href='url'>another link</a>"
var c = "get the file <a href='url'>last link</a>"

using
document.getElementById("myspan").textContent=a;
on
<span id="myspan">first text</span>

with a timer but it just shows the reference target as text not runing the code, even tho it does shows correctly on the source code. If the jquery approch is not really a solution, the use of:

document.getElementById("myspan").innerHTML = a to c;
is the best way to make it work.

Many people still come across this question (in 2022) and the available answers are not really up to date.

Use innerText is the best method

As you can see in the MDM Docs innerText is the best way to retrieve and change the text of a <span> HTML element via Javascript.

The innerText property is VERY well supported (97.53% of all web users according to Caniuse)

How to use

Simple retrieve and set new text with the property like this:

let mySpan = document.getElementById("myspan");

console.log(mySpan.innerText);

mySpan.innerText = "Setting a new text content into the span element.";

Why better than innerHTML ?

Don't use innerHTML to updating the content with user inputs, this can lead to major vulnerability since the string content you will set will be interpreted and converted into HTML tags.

This means users can insert script(s) into your site, this is known as XSS attacks/vulnerabilities (Cross-site scripting).

Why better than textContent ?

First point textContent isn't supported by IE8 (but I think in 2022 nobody cares anymore). But the main element is the true difference of result you can get using textContent instead of innerText.

The example from the MDM documentation is perfect to illustrate that, so we have the following setup:

<p id="source">
  <style>#source { color: red;  } #text { text-transform: uppercase; }</style>
<span id=text>Take a look at<br>how this text<br>is interpreted
       below.</span>
  <span style="display:none">HIDDEN TEXT</span>
</p>

If you use innerText to retrieve the text content of <p id="source"> we get:

TAKE A LOOK AT
HOW THIS TEXT
IS INTERPRETED BELOW.

This is perfectly what we wanted.

Now using textContent we get:

  #source { color: red;  } #text { text-transform: uppercase; }
Take a look athow this textis interpreted
       below.
  HIDDEN TEXT

Not exactly what you expected...

This is why using textContent isn't the correct way.

Last point

If you goal is only to append text to a <p> or <span> HTML element, the answer from nicooo. is right you can create a new text node and append it to you existing element like this:

let mySpan = document.getElementById("myspan");

const newTextNode = document.createTextNode("Youhou!"),

mySpan.appendChild(newTextNode);

(function ($) {
    $(document).ready(function(){
    $("#myspan").text("This is span");
  });
}(jQuery));

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<span id="myspan"> hereismytext </span>

user text() to change span text.

I used this one document.querySelector('ElementClass').innerText = 'newtext';

Appears to work with span, texts within classes/buttons

For some reason, it seems that using "text" attribute is the way to go with most browsers. It worked for me

$("#span_id").text("text value to assign");