Python MySQLdb connection - when to open/close new connections?

Question

I'm using a few apps running Tornado Web server which all connect to a MySql DB using mysqldb. When I spin up the server, it instantiates a DB class (below) which opens a connection to the DB. All transactions are made using this same connection - which I'm not sure is a good idea.

class RDSdb(object):

    def __init__(self):
        self.connect()

    def connect(self):
        self.connection = MySQLdb.connect(cursorclass = MySQLdb.cursors.SSDictCursor, host=self.RDS_HOST,
                                    user=self.RDS_USER, passwd=self.RDS_PASS, db=self.RDS_DB)

    def get_cursor(self):
        try:
            cursor = self.connection.cursor()
        except (AttributeError, MySQLdb.OperationalError):
            self.connect()
            cursor = self.connection.cursor()
        return cursor

    def fetch_by_query(self, query):
        cursor = self.get_cursor()
        cursor.execute(query)
        result = cursor.fetchall()
        cursor.close()
        return result

I'm pretty sure I shouldn't open/close a new connection for every transaction, but then, when should I?

I noticed something else that's a bit off, which I'm certain is related : when I need to update one of my db table's schema (ex : alter table), the whole table in question gets locked and unresponsive - until I kill my 3 apps with open connections to the DB - I realize that one of those connections was holding up this update.

Best practices when it comes to this? Ideas?

thanks.