Open directory dialog

You can use the built-in FolderBrowserDialog class for this. Don't mind that it's in the System.Windows.Forms namespace.

using (var dialog = new System.Windows.Forms.FolderBrowserDialog())
{
    System.Windows.Forms.DialogResult result = dialog.ShowDialog();
}

If you want the window to be modal over some WPF window, see the question How to use a FolderBrowserDialog from a WPF application.

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EDIT: If you want something a bit more fancy than the plain, ugly Windows Forms FolderBrowserDialog, there are some alternatives that allow you to use the Vista dialog instead:

• Third-party libraries, such as Ookii dialogs (.NET 4.5+)

• The Windows API Code Pack-Shell:

    using Microsoft.WindowsAPICodePack.Dialogs;
  
    ...
  
    var dialog = new CommonOpenFileDialog();
    dialog.IsFolderPicker = true;
    CommonFileDialogResult result = dialog.ShowDialog();
  

  Note that this dialog is not available on operating systems older than Windows Vista, so be sure to check CommonFileDialog.IsPlatformSupported first.

I created a UserControl which is used like this:

  <UtilitiesWPF:FolderEntry Text="{Binding Path=LogFolder}" Description="Folder for log files"/>

The xaml source looks like this:

<UserControl x:Class="Utilities.WPF.FolderEntry"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml">
    <DockPanel>
        <Button Margin="0" Padding="0" DockPanel.Dock="Right" Width="Auto" Click="BrowseFolder">...</Button>
        <TextBox Height="Auto" HorizontalAlignment="Stretch" DockPanel.Dock="Right" 
           Text="{Binding Text, RelativeSource={RelativeSource FindAncestor, AncestorType={x:Type UserControl}}}" />
    </DockPanel>
</UserControl>

and the code-behind

public partial class FolderEntry {
    public static DependencyProperty TextProperty = DependencyProperty.Register("Text", typeof(string), typeof(FolderEntry), new FrameworkPropertyMetadata(null, FrameworkPropertyMetadataOptions.BindsTwoWayByDefault));
    public static DependencyProperty DescriptionProperty = DependencyProperty.Register("Description", typeof(string), typeof(FolderEntry), new PropertyMetadata(null));

    public string Text { get { return GetValue(TextProperty) as string; } set { SetValue(TextProperty, value); }}

    public string Description { get { return GetValue(DescriptionProperty) as string; } set { SetValue(DescriptionProperty, value); } }

    public FolderEntry() { InitializeComponent(); }

    private void BrowseFolder(object sender, RoutedEventArgs e) {
        using (FolderBrowserDialog dlg = new FolderBrowserDialog()) {
            dlg.Description = Description;
            dlg.SelectedPath = Text;
            dlg.ShowNewFolderButton = true;
            DialogResult result = dlg.ShowDialog();
            if (result == System.Windows.Forms.DialogResult.OK) {
                Text = dlg.SelectedPath;
                BindingExpression be = GetBindingExpression(TextProperty);
                if (be != null)
                    be.UpdateSource();
            }
        }
    }
 }

Ookii folder dialog can be found at Nuget.

PM> Install-Package Ookii.Dialogs.Wpf

And, example code is as below.

var dialog = new Ookii.Dialogs.Wpf.VistaFolderBrowserDialog();
if (dialog.ShowDialog(this).GetValueOrDefault())
{
    textBoxFolderPath.Text = dialog.SelectedPath;
}

More information on how to use it: https://github.com/augustoproiete/ookii-dialogs-wpf