Using the getDay method of Date objects, you can know the number of day of the week (being 0=Sunday, 1=Monday, etc).

You can then subtract that number of days plus one, for example:

function getMonday(d) {
  d = new Date(d);
  var day = d.getDay(),
    diff = d.getDate() - day + (day == 0 ? -6 : 1); // adjust when day is sunday
  return new Date(d.setDate(diff));
}

console.log( getMonday(new Date()) ); // e.g. Mon Nov 08 2010

Your code DOES get the previous Monday on a Sunday

What does "If the week starts on a Sunday" mean to your?

In JS Sunday is the 0th or first day

// Get the monday date of current week
function getMondayOfCurrentWeek(d) {
  const date = new Date(d);
  const day = date.getDay(); // Sunday - Saturday : 0 - 6
  //  Day of month - day of week (-6 if Sunday), otherwise +1
  const diff = date.getDate() - day + (day === 0 ? -6 : 1);
  return date.setDate(diff), date;
}

console.log(getMondayOfCurrentWeek(new Date()).toDateString() )

console.log(getMondayOfCurrentWeek(new Date(2022,3,3,15,0,0,0)).toDateString() ); // Sunday 3rd of April 2022

var curr = new Date; // get current date
var first = curr.getDate() - curr.getDay(); // First day is the day of the month - the day of the week
var last = first + 6; // last day is the first day + 6

var firstday = new Date(curr.setDate(first)).toUTCString();
var lastday = new Date(curr.setDate(last)).toUTCString();

firstday
"Sun, 06 Mar 2011 12:25:40 GMT"
lastday
"Sat, 12 Mar 2011 12:25:40 GMT"

This works for firstday = sunday of this week and last day = saturday for this week. Extending it to run Monday to sunday is trivial.

Making it work with first and last days in different months is left as an exercise for the user

<script>
Date.prototype.getWeek = function(start)
{
        //Calcing the starting point
    start = start || 0;
    var today = new Date(this.setHours(0, 0, 0, 0));
    var day = today.getDay() - start;
    var date = today.getDate() - day;

        // Grabbing Start/End Dates
    var StartDate = new Date(today.setDate(date));
    var EndDate = new Date(today.setDate(date + 6));
    return [StartDate, EndDate];
}

// test code
var Dates = new Date().getWeek();
alert(Dates[0].toLocaleDateString() + ' to '+ Dates[1].toLocaleDateString())
</script>

I believe this works.

You can get the previous Monday by getting the Monday of this week and subtracting 7 days. The Sunday will be one day before that, so:

var d = new Date();

// set to Monday of this week
d.setDate(d.getDate() - (d.getDay() + 6) % 7);

// set to previous Monday
d.setDate(d.getDate() - 7);

// create new date of day before
var sunday = new Date(d.getFullYear(), d.getMonth(), d.getDate() - 1);

For 2012-12-03 I get:

Mon 26 Nov 2012
Sun 25 Nov 2012

Is that what you want?

// Or new date for the following Sunday
var sunday = new Date(d.getFullYear(), d.getMonth(), d.getDate() + 6);

which gives

Sun 02 Dec 2012

In general, you can manipulate date objects by add and subtracting years, months and days. The object will handle negative values automatically, e.g.

var d = new Date(2012,11,0)

Will create a date for 2012-11-30 (noting that months are zero based so 11 is December). Also:

d.setMonth(d.getMonth() - 1); // 2012-10-30

d.setDate(d.getDate() - 30);  // 2012-09-30

var prevMonday = new Date();
prevMonday.setDate(prevMonday.getDate() - (prevMonday.getDay() + 6) % 7);

I am a little late but I like this solution. It's a one liner and not that complicated. Making a function for that seems overkill to me.