How can I get the DateTime for the start of the week?

Use an extension method. They're the answer to everything, you know! ;)

public static class DateTimeExtensions
{
    public static DateTime StartOfWeek(this DateTime dt, DayOfWeek startOfWeek)
    {
        int diff = (7 + (dt.DayOfWeek - startOfWeek)) % 7;
        return dt.AddDays(-1 * diff).Date;
    }
}

Which can be used as follows:

DateTime dt = DateTime.Now.StartOfWeek(DayOfWeek.Monday);
DateTime dt = DateTime.Now.StartOfWeek(DayOfWeek.Sunday);

A little more verbose and culture-aware:

System.Globalization.CultureInfo ci = 
    System.Threading.Thread.CurrentThread.CurrentCulture;
DayOfWeek fdow = ci.DateTimeFormat.FirstDayOfWeek;
DayOfWeek today = DateTime.Now.DayOfWeek;
DateTime sow = DateTime.Now.AddDays(-(today - fdow)).Date;

Let's combine the culture-safe answer and the extension method answer:

public static class DateTimeExtensions
{
    public static DateTime StartOfWeek(this DateTime dt, DayOfWeek startOfWeek)
    {
        System.Globalization.CultureInfo ci = System.Threading.Thread.CurrentThread.CurrentCulture;
        DayOfWeek fdow = ci.DateTimeFormat.FirstDayOfWeek;
        return DateTime.Today.AddDays(-(DateTime.Today.DayOfWeek- fdow));
    }
}

This would give you the preceding Sunday (I think):

DateTime t = DateTime.Now;
t -= new TimeSpan ((int) t.DayOfWeek, 0, 0, 0);

For Monday

DateTime startAtMonday = DateTime.Now.AddDays(DayOfWeek.Monday - DateTime.Now.DayOfWeek);

For Sunday

DateTime startAtSunday = DateTime.Now.AddDays(DayOfWeek.Sunday- DateTime.Now.DayOfWeek);

This may be a bit of a hack, but you can cast the .DayOfWeek property to an int (it's an enum and since its not had its underlying data type changed it defaults to int) and use that to determine the previous start of the week.

It appears the week specified in the DayOfWeek enum starts on Sunday, so if we subtract 1 from this value that'll be equal to how many days the Monday is before the current date. We also need to map the Sunday (0) to equal 7 so given 1 - 7 = -6 the Sunday will map to the previous Monday:-

DateTime now = DateTime.Now;
int dayOfWeek = (int)now.DayOfWeek;
dayOfWeek = dayOfWeek == 0 ? 7 : dayOfWeek;
DateTime startOfWeek = now.AddDays(1 - (int)now.DayOfWeek);

The code for the previous Sunday is simpler as we don't have to make this adjustment:-

DateTime now = DateTime.Now;
int dayOfWeek = (int)now.DayOfWeek;
DateTime startOfWeek = now.AddDays(-(int)now.DayOfWeek);

Step 1: Create a static class

  public static class TIMEE
{
    public static DateTime StartOfWeek(this DateTime dt, DayOfWeek startOfWeek)
    {
        int diff = (7 + (dt.DayOfWeek - startOfWeek)) % 7;
        return dt.AddDays(-1 * diff).Date;
    }
    public static DateTime EndOfWeek(this DateTime dt, DayOfWeek startOfWeek)
    {
        int diff = (7 - (dt.DayOfWeek - startOfWeek)) % 7;
        return dt.AddDays(1 * diff).Date;
    }
}

Step2: Use this class to get both start and end day of the week

 DateTime dt =TIMEE.StartOfWeek(DateTime.Now ,DayOfWeek.Monday);
        DateTime dt1 = TIMEE.EndOfWeek(DateTime.Now, DayOfWeek.Sunday);

This would give you midnight on the first Sunday of the week:

DateTime t = DateTime.Now;
t -= new TimeSpan ((int) t.DayOfWeek, t.Hour, t.Minute, t.Second);

This gives you the first Monday at midnight:

DateTime t = DateTime.Now;
t -= new TimeSpan ((int) t.DayOfWeek - 1, t.Hour, t.Minute, t.Second);

Did it for Monday but similar logic for Sunday.

public static DateTime GetStartOfWeekDate()
{
    // Get today's date
    DateTime today = DateTime.Today;
    // Get the value for today. DayOfWeek is an enum with 0 being Sunday, 1 Monday, etc
    var todayDayOfWeek = (int)today.DayOfWeek;

    var dateStartOfWeek = today;
    // If today is not Monday, then get the date for Monday
    if (todayDayOfWeek != 1)
    {
        // How many days to get back to Monday from today
        var daysToStartOfWeek = (todayDayOfWeek - 1);
        // Subtract from today's date the number of days to get to Monday
        dateStartOfWeek = today.AddDays(-daysToStartOfWeek);
    }

    return dateStartOfWeek;

}

The following method should return the DateTime that you want. Pass in true for Sunday being the first day of the week, false for Monday:

private DateTime getStartOfWeek(bool useSunday)
{
    DateTime now = DateTime.Now;
    int dayOfWeek = (int)now.DayOfWeek;

    if(!useSunday)
        dayOfWeek--;

    if(dayOfWeek < 0)
    {// day of week is Sunday and we want to use Monday as the start of the week
    // Sunday is now the seventh day of the week
        dayOfWeek = 6;
    }

    return now.AddDays(-1 * (double)dayOfWeek);
}

You could use the excellent Umbrella library:

using nVentive.Umbrella.Extensions.Calendar;
DateTime beginning = DateTime.Now.BeginningOfWeek();

However, they do seem to have stored Monday as the first day of the week (see the property nVentive.Umbrella.Extensions.Calendar.DefaultDateTimeCalendarExtensions.WeekBeginsOn), so that previous localized solution is a bit better. Unfortunate.

Edit: looking closer at the question, it looks like Umbrella might actually work for that too:

// Or DateTime.Now.PreviousDay(DayOfWeek.Monday)
DateTime monday = DateTime.Now.PreviousMonday(); 
DateTime sunday = DateTime.Now.PreviousSunday();

Although it's worth noting that if you ask for the previous Monday on a Monday, it'll give you seven days back. But this is also true if you use BeginningOfWeek, which seems like a bug :(.

Following on from Compile This' Answer, use the following method to obtain the date for any day of the week:

public static DateTime GetDayOfWeek(DateTime dateTime, DayOfWeek dayOfWeek)
{
   var monday = dateTime.Date.AddDays((7 + (dateTime.DayOfWeek - DayOfWeek.Monday) % 7) * -1);

   var diff = dayOfWeek - DayOfWeek.Monday;

   if (diff == -1)
   {
      diff = 6;
   }

   return monday.AddDays(diff);
} 

Here is a combination of a few of the answers. It uses an extension method that allows the culture to be passed in, if one is not passed in, the current culture is used. This will give it max flexibility and re-use.

/// <summary>
/// Gets the date of the first day of the week for the date.
/// </summary>
/// <param name="date">The date to be used</param>
/// <param name="cultureInfo">If none is provided, the current culture is used</param>
/// <returns>The date of the beggining of the week based on the culture specifed</returns>
public static DateTime StartOfWeek(this DateTime date, CultureInfo cultureInfo=null) =>         
             date.AddDays(-1 * (7 + (date.DayOfWeek - (cultureInfo??CultureInfo.CurrentCulture).DateTimeFormat.FirstDayOfWeek)) % 7).Date;

Example Usage:

public static void TestFirstDayOfWeekExtension() {          
        DateTime date = DateTime.Now;
        foreach(System.Globalization.CultureInfo culture in CultureInfo.GetCultures(CultureTypes.UserCustomCulture | CultureTypes.SpecificCultures)) {
            Console.WriteLine($"{culture.EnglishName}: {date.ToShortDateString()} First Day of week: {date.StartOfWeek(culture).ToShortDateString()}");
        }
    }

if you want saturday or sunday or any day of week but not exceeding current week(Sat-Sun) I got you covered with this piece of code.

public static DateTime GetDateInCurrentWeek(this DateTime date, DayOfWeek day)
{
    var temp = date;
    var limit = (int)date.DayOfWeek;
    var returnDate = DateTime.MinValue;

    if (date.DayOfWeek == day) return date;

    for (int i = limit; i < 6; i++)
    {
        temp = temp.AddDays(1);

        if (day == temp.DayOfWeek)
        {
            returnDate = temp;
            break;
        }
    }
    if (returnDate == DateTime.MinValue)
    {
        for (int i = limit; i > -1; i++)
        {
            date = date.AddDays(-1);

            if (day == date.DayOfWeek)
            {
                returnDate = date;
                break;
            }
        }
    }
    return returnDate;
}

We like one-liners : Get the difference between the current culture's first day of week and the current day then subtract the number of days from the current day

var weekStartDate = DateTime.Now.AddDays(-((int)now.DayOfWeek - (int)DateTimeFormatInfo.CurrentInfo.FirstDayOfWeek));

Calculating this way lets you choose which day of the week indicates the start of a new week (in the example I chose Monday).

Note that doing this calculation for a day that is a Monday will give the current Monday and not the previous one.

//Replace with whatever input date you want
DateTime inputDate = DateTime.Now;

//For this example, weeks start on Monday
int startOfWeek = (int)DayOfWeek.Monday;

//Calculate the number of days it has been since the start of the week
int daysSinceStartOfWeek = ((int)inputDate.DayOfWeek + 7 - startOfWeek) % 7;

DateTime previousStartOfWeek = inputDate.AddDays(-daysSinceStartOfWeek);

I work with a lot of schools, so correctly using Monday as the first day of the week is important here.

A lot of the most terse answers here don't work on Sunday -- we often end up returning the date of tomorrow on Sunday, which is not good for running a report on last week's activities.

Here's my solution, which returns last Monday on Sunday, and today on Monday.

// Adding 7 so remainder is always positive; Otherwise % returns -1 on Sunday.
var daysToSubtract = (7 + (int)today.DayOfWeek - (int)DayOfWeek.Monday) % 7;

var monday = today
    .AddDays(-daysToSubtract)
    .Date;

Remember to use a method parameter for "today" so it's unit testable!!

Try to create a function which uses recursion. Your DateTime object is an input and function returns a new DateTime object which stands for the beginning of the week.

    DateTime WeekBeginning(DateTime input)
    {
        do
        {
            if (input.DayOfWeek.ToString() == "Monday")
                return input;
            else
                return WeekBeginning(input.AddDays(-1));
        } while (input.DayOfWeek.ToString() == "Monday");
    }

I did it like this:

DateTime.Now.Date.AddDays(-(DateTime.Now.Date.DayOfWeek == 0 ? 7 : (int)DateTime.Now.Date.DayOfWeek) + 1)

All this code does is subtracting a number of days from the given datetime.

If day of week is 0(sunday) then subtract 7 else subtract the day of the week.

Then add 1 day to the result of the previous line, which gives you the monday of that date.

This way you can play around with the number(1) at the end to get the desired day.

private static DateTime GetDay(DateTime date, int daysAmount = 1)
{
    return date.Date.AddDays(-(date.Date.DayOfWeek == 0 ? 7 : (int)date.Date.DayOfWeek) + daysAmount);
}

If you really want to use the DayOfWeek enum then something like this can be used... though I presonally prefer the above one, as I can add or subtract any amount of days.

private static DateTime GetDayOfWeek(DateTime date, DayOfWeek dayOfWeek = DayOfWeek.Monday)
{
    return date.Date.AddDays(-(date.Date.DayOfWeek == 0 ? 7 : (int)date.Date.DayOfWeek) + (dayOfWeek == 0 ? 7 : (int)dayOfWeek));
}