diff --git a/.travis.yml b/.travis.yml
index 4b4ea79491..16f47692ff 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -27,8 +27,8 @@ sudo: false
 #   * stage_osx
 stage_generic: &stage_generic
   install:
-    - pip install -U 'pip<20.0.0'
-    - pip install cython
+    - pip install -U pip
+    - pip install -U cython packaging
     # Install terra from master on aer master else use the most recent release
     - |
       if [[ $TRAVIS_BRANCH == "master" ]] ; then
@@ -149,7 +149,7 @@ jobs:
         - docker
       env:
         - CIBW_BEFORE_BUILD="pip install -U Cython pip virtualenv pybind11 && yum install -y openblas-devel"
-        - CIBW_SKIP="cp27-* cp34-* *-manylinux_i686 pp*"
+        - CIBW_SKIP="cp27-* cp34-* cp35-* *-manylinux_i686 pp*"
         - CIBW_MANYLINUX_X86_64_IMAGE="manylinux2010"
         - CIBW_MANYLINUX_I686_IMAGE="manylinux2010"
         - CIBW_TEST_COMMAND="python3 {project}/tools/verify_wheels.py"
@@ -223,13 +223,6 @@ jobs:
       <<: *stage_linux
       python: 3.6
 
-    # GNU/Linux, Python 3.5
-    - stage: test
-      if: type = cron
-      name: Python 3.5 Tests Linux
-      <<: *stage_linux
-      python: 3.5
-
     # MacOS, Python 3.8 (via pyenv)
     - stage: test
       if: type = cron
@@ -261,16 +254,6 @@ jobs:
         - PYTHON_VERSION=3.6.5
         - QISKIT_SUPPRESS_PACKAGING_WARNINGS=Y
 
-    # MacOS, Python 3.5 (via pyenv)
-    - stage: test
-      if: type = cron
-      name: Python 3.5 Tests MacOS
-      <<: *stage_osx
-      python: 3.5
-      env:
-        - MPLBACKEND=ps
-        - PYTHON_VERSION=3.5.6
-        - QISKIT_SUPPRESS_PACKAGING_WARNINGS=Y
     # "deploy" stage
     ##########################################################################
     #
@@ -284,7 +267,7 @@ jobs:
       env:
         - CIBW_BEFORE_ALL="yum install -y openblas-devel"
         - CIBW_BEFORE_BUILD="pip install -U Cython pip virtualenv pybind11"
-        - CIBW_SKIP="cp27-* cp34-* *-manylinux_i686 pp*"
+        - CIBW_SKIP="cp27-* cp34-* cp35-* *-manylinux_i686 pp*"
         - CIBW_MANYLINUX_X86_64_IMAGE="manylinux2010"
         - CIBW_MANYLINUX_I686_IMAGE="manylinux2010"
         - TWINE_USERNAME=qiskit
@@ -315,7 +298,7 @@ jobs:
       if: tag IS present
       env:
         - CIBW_BEFORE_BUILD="pip install -U Cython pip virtualenv pybind11"
-        - CIBW_SKIP="cp27-* cp34-* pp*"
+        - CIBW_SKIP="cp27-* cp34-* cp35-* pp*"
         - TWINE_USERNAME=qiskit
         - CIBW_TEST_COMMAND="python3 {project}/tools/verify_wheels.py"
         - CIBW_TEST_REQUIRES="git+https://github.com/Qiskit/qiskit-terra.git"
diff --git a/azure-pipelines.yml b/azure-pipelines.yml
index b4694964cf..cf3f8effdc 100644
--- a/azure-pipelines.yml
+++ b/azure-pipelines.yml
@@ -33,7 +33,7 @@ stages:
         - bash: |
             set -x
             set -e
-            for version in 3.5 3.6 3.7 3.8 ; do
+            for version in 3.6 3.7 3.8 ; do
                 conda create --yes --quiet --name qiskit-aer-$version python=$version
             done
           displayName: Create Anaconda environments
@@ -41,7 +41,7 @@ stages:
             set -x
             set -e
             mkdir wheelhouse
-            for version in 3.5 3.6 3.7 3.8 ; do
+            for version in 3.6 3.7 3.8 ; do
                 source activate qiskit-aer-$version
                 conda update --yes -n base conda
                 conda config --add channels conda-forge
@@ -76,8 +76,6 @@ stages:
         strategy:
           maxParallel: 2
           matrix:
-            Python35:
-              python.version: '3.5'
             Python36:
               python.version: '3.6'
             Python37:
@@ -109,12 +107,6 @@ stages:
               conda config --add channels conda-forge
               conda install --yes --quiet --name qiskit-aer python=$(python.version) numpy cmake virtualenv pip setuptools pybind11 cython scipy
             displayName: Create Anaconda environments
-          - bash: |
-              set -e
-              source activate qiskit-aer
-              pip install -U -c constraints.txt numpy scipy
-            displayName: "Install 3.5 constrained dependencies"
-            condition: eq(variables['python.version'], '3.5')
           - bash: |
               set -x
               set -e
@@ -259,8 +251,6 @@ stages:
         strategy:
           maxParallel: 3
           matrix:
-            Python35:
-              python.version: '3.5'
             Python36:
               python.version: '3.6'
             Python37:
@@ -313,18 +303,6 @@ stages:
               fi
             displayName: "Install dependencies py3.8"
             condition: eq(variables['python.version'], '3.8')
-          - bash: |
-              set -e
-              source activate qiskit-aer-$(Build.BuildNumber)
-              pip install -U pip setuptools wheel virtualenv
-              pip install -c constraints.txt -r requirements-dev.txt
-              if [[ $SYSTEM_PULLREQUEST_TARGETBRANCH == *"master"* || $BUILD_SOURCEBRANCH == *"master"* ]] ; then
-                  pip install git+https://github.com/Qiskit/qiskit-terra.git
-              else
-                  pip install qiskit-terra
-              fi
-            displayName: "Install dependencies"
-            condition: eq(variables['python.version'], '3.5')
           - bash: |
               set -e
               source activate qiskit-aer-$(Build.BuildNumber)
diff --git a/releasenotes/notes/drop-py35-814876e9d7354a39.yaml b/releasenotes/notes/drop-py35-814876e9d7354a39.yaml
new file mode 100644
index 0000000000..e30bc0e281
--- /dev/null
+++ b/releasenotes/notes/drop-py35-814876e9d7354a39.yaml
@@ -0,0 +1,7 @@
+---
+upgrade:
+  - |
+    The deprecated support for running qiskit-aer with Python 3.5 has
+    been removed. To use qiskit-aer >=0.7.0 you will now need at
+    least Python 3.6. If you are using Python 3.5 the last version which will
+    work is qiskit-aer 0.6.x.
diff --git a/requirements-dev.txt b/requirements-dev.txt
index c3fb3ce1be..57fce72ada 100644
--- a/requirements-dev.txt
+++ b/requirements-dev.txt
@@ -12,4 +12,4 @@ sphinx-rtd-theme>=0.4.0
 sphinx-tabs>=1.1.11
 sphinx-automodapi
 jupyter-sphinx;python_version<'3.8'
-reno>=3.1.0;python_version>'3.5'
+reno>=3.1.0
diff --git a/setup.py b/setup.py
index fd71e9f58c..c1c050e679 100644
--- a/setup.py
+++ b/setup.py
@@ -33,10 +33,8 @@
 # also build time/setup requirements and will be added to both lists
 # of requirements
 common_requirements = [
-    'numpy>=1.16.3;python_version>"3.5"',
-    'numpy>=1.16.3,<1.19.0;python_version<"3.6"',
-    'scipy>=1.0;python_version>"3.5"',
-    'scipy>=1.0,<1.5.0;python_version<"3.6"',
+    'numpy>=1.16.3',
+    'scipy>=1.0',
     'cython>=0.27.1',
     'pybind11>=2.4'  # This isn't really an install requirement,
                      # Pybind11 is required to be pre-installed for
@@ -92,12 +90,12 @@
         "Operating System :: POSIX :: Linux",
         "Programming Language :: C++",
         "Programming Language :: Python :: 3 :: Only",
-        "Programming Language :: Python :: 3.5",
         "Programming Language :: Python :: 3.6",
         "Programming Language :: Python :: 3.7",
         "Programming Language :: Python :: 3.8",
         "Topic :: Scientific/Engineering",
     ],
+    python_requires=">=3.6",
     install_requires=requirements,
     setup_requires=setup_requirements,
     include_package_data=True,
diff --git a/tox.ini b/tox.ini
index 03b7386894..96f8d7dcb5 100644
--- a/tox.ini
+++ b/tox.ini
@@ -1,6 +1,6 @@
 [tox]
 minversion = 2.1
-envlist = py35, py36, py37, lint
+envlist = py36, py37, py38, lint
 skipsdist = True
 
 [testenv]