On day 21 June in the north hemisphere will begin the summer

www.pierpaoloricci.it/dati/estate_eng.htm

I suppose technically.

But I consider Summer to start 1st June and end 31st August.

Of course weatherwise, we've had it. a couple of weeks of sun and now clouds. 

Indeed, today is the beginning of “astronomical” summer.  The US Weather Service concurs with Oliver and considers June, July and August to comprise summer for record keeping purposes.  In much of the US the warmest three months of the year averages from about June 5 to about September 5.  Nevertheless, culturally many Americans consider summer to begin on Memorial Day (last Monday in May) and to end on Labor Day (first Monday in September).  To most people summer is more a state of mind than an astronomical event.  But the dates of solstices and equinoxes are of interest, and certainly were so in antiquity when they were major holidays.

I should add that for the ancients there were no words for spring or autumn.  Summer was considered to have begun on May Day and winter on Halloween.  It was also the custom to refer to the summer solstice as Midsummer Day.