While the accepted answer can be considered valid, it has some issues.

• using setGeometry() is not appropriate (and the reason for using it was wrong) since it doesn’t consider possible frame margins set by the style;
• the position computation is unnecessarily complex;
• resizing the title bar to the total width is wrong, since it doesn’t consider the buttons and can also cause recursion problems in certain situations (like not setting the minimum size of the main window); also, if the title is too big, it makes impossible to resize the main window;
• buttons should not accept focus;
• setting a layout creates a restraint for the "main widget" or layout, so the title should not be added, but the contents margins of the widget should be used instead;

I revised the code to provide a better base for the main window, simplify the moving code, and add other features like the Qt windowTitle() property support, standard QStyle icons for buttons (instead of text), and proper maximize/normal button icons. Note that the title label is not added to the layout.

class MainWindow(QWidget):
    def __init__(self):
        super(MainWindow, self).__init__()
        self.setWindowFlags(self.windowFlags() | Qt.FramelessWindowHint)

        self.titleBar = MyBar(self)
        self.setContentsMargins(0, self.titleBar.height(), 0, 0)

        self.resize(640, self.titleBar.height() + 480)

    def changeEvent(self, event):
        if event.type() == event.WindowStateChange:
            self.titleBar.windowStateChanged(self.windowState())

    def resizeEvent(self, event):
        self.titleBar.resize(self.width(), self.titleBar.height())


class MyBar(QWidget):
    clickPos = None
    def __init__(self, parent):
        super(MyBar, self).__init__(parent)
        self.setAutoFillBackground(True)
        
        self.setBackgroundRole(QPalette.Shadow)
        # alternatively:
        # palette = self.palette()
        # palette.setColor(palette.Window, Qt.black)
        # palette.setColor(palette.WindowText, Qt.white)
        # self.setPalette(palette)

        layout = QHBoxLayout(self)
        layout.setContentsMargins(1, 1, 1, 1)
        layout.addStretch()

        self.title = QLabel("My Own Bar", self, alignment=Qt.AlignCenter)
        # if setPalette() was used above, this is not required
        self.title.setForegroundRole(QPalette.Light)

        style = self.style()
        ref_size = self.fontMetrics().height()
        ref_size += style.pixelMetric(style.PM_ButtonMargin) * 2
        self.setMaximumHeight(ref_size + 2)

        btn_size = QSize(ref_size, ref_size)
        for target in ('min', 'normal', 'max', 'close'):
            btn = QToolButton(self, focusPolicy=Qt.NoFocus)
            layout.addWidget(btn)
            btn.setFixedSize(btn_size)

            iconType = getattr(style, 
                'SP_TitleBar{}Button'.format(target.capitalize()))
            btn.setIcon(style.standardIcon(iconType))

            if target == 'close':
                colorNormal = 'red'
                colorHover = 'orangered'
            else:
                colorNormal = 'palette(mid)'
                colorHover = 'palette(light)'
            btn.setStyleSheet('''
                QToolButton {{
                    background-color: {};
                }}
                QToolButton:hover {{
                    background-color: {}
                }}
            '''.format(colorNormal, colorHover))

            signal = getattr(self, target + 'Clicked')
            btn.clicked.connect(signal)

            setattr(self, target + 'Button', btn)

        self.normalButton.hide()

        self.updateTitle(parent.windowTitle())
        parent.windowTitleChanged.connect(self.updateTitle)

    def updateTitle(self, title=None):
        if title is None:
            title = self.window().windowTitle()
        width = self.title.width()
        width -= self.style().pixelMetric(QStyle.PM_LayoutHorizontalSpacing) * 2
        self.title.setText(self.fontMetrics().elidedText(
            title, Qt.ElideRight, width))

    def windowStateChanged(self, state):
        self.normalButton.setVisible(state == Qt.WindowMaximized)
        self.maxButton.setVisible(state != Qt.WindowMaximized)

    def mousePressEvent(self, event):
        if event.button() == Qt.LeftButton:
            self.clickPos = event.windowPos().toPoint()

    def mouseMoveEvent(self, event):
        if self.clickPos is not None:
            self.window().move(event.globalPos() - self.clickPos)

    def mouseReleaseEvent(self, QMouseEvent):
        self.clickPos = None

    def closeClicked(self):
        self.window().close()

    def maxClicked(self):
        self.window().showMaximized()

    def normalClicked(self):
        self.window().showNormal()

    def minClicked(self):
        self.window().showMinimized()

    def resizeEvent(self, event):
        self.title.resize(self.minButton.x(), self.height())
        self.updateTitle()


if __name__ == "__main__":
    app = QApplication(sys.argv)
    mw = MainWindow()
    layout = QVBoxLayout(mw)
    widget = QTextEdit()
    layout.addWidget(widget)
    mw.show()
    mw.setWindowTitle('My custom window with a very, very long title')
    sys.exit(app.exec_())

This is for the people who are going to implement custom title bar in PyQt6 or PySide6

The below changes should be done in the answer given by @musicamante

    def mousePressEvent(self, event):
        if event.button() == Qt.LeftButton:
            # self.clickPos = event.windowPos().toPoint()
            self.clickPos = event.scenePosition().toPoint()

    def mouseMoveEvent(self, event):
        if self.clickPos is not None:
            # self.window().move(event.globalPos() - self.clickPos)
            self.window().move(event.globalPosition().toPoint() - self.clickPos)

if __name__ == "__main__":
    app = QApplication(sys.argv)
    mw = MainWindow()
    mw.show()
    # sys.exit(app.exec_())
    sys.exit(app.exec())

References:
QMouseEvent.globalPosition(),
QMouseEvent.scenePosition()

This method of moving Windows with Custom Widget doesn’t work with WAYLAND. If anybody has a solution for that please post it here for future reference

Working functions for WAYLAND and PyQT6/PySide6 :

def mousePressEvent(self, event):
    if event.button() == Qt.MouseButton.LeftButton:
        self._move()
        return super().mousePressEvent(event)

def _move(self):
    window = self.window().windowHandle()
    window.startSystemMove()

Please check.