Each of the six members from the Skateer University hockey team shakes hands with each of the six members of the Iceburg Tech team. How many handshakes take place?
Each member of the team from Skateer University will shake hands once with each member of the Iceburg Tech team, which will be 6 handshakes each.  This will give a total of $6\cdot 6=\boxed{36}$ handshakes.