Gravitational-wave astronomy has been firmly established with the detection of gravitational waves from the merger of ten stellar-mass binary black holes and a neutron star binary. This paper reports on the all-sky search for gravitational waves from intermediate mass black hole binaries in the first and second observing runs of the Advanced LIGO and Virgo network. The search uses three independent algorithms: two based on matched filtering of the data with waveform templates of gravitational-wave signals from compact binaries, and a third, model-independent algorithm that employs no signal model for the incoming signal. No intermediate mass black hole binary event is detected in this search. Consequently, we place upper limits on the merger rate density for a family of intermediate mass black hole binaries. In particular, we choose sources with total masses
M
=
m
1
+
m
2
∈
[
120
,
800
]
 
 
M
⊙
and mass ratios
q
=
m
2
/
m
1
∈
[
0.1
,
1.0
]
. For the first time, this calculation is done using numerical relativity waveforms (which include higher modes) as models of the real emitted signal. We place a most stringent upper limit of
0.20
 
 
Gpc
−
3
 
yr
−
1
(in comoving units at the 90% confidence level) for equal-mass binaries with individual masses
m
1
,
2
=
100
 
 
M
⊙
and dimensionless spins
χ
1
,
2
=
0.8
aligned with the orbital angular momentum of the binary. This improves by a factor of
∼
5
that reported after Advanced LIGO’s first observing run.