diff --git a/ci/etc/valgrind-cran.supp b/ci/etc/valgrind-cran.supp
new file mode 100644
index 000000000000..4d2922026082
--- /dev/null
+++ b/ci/etc/valgrind-cran.supp
@@ -0,0 +1,34 @@
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements.  See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership.  The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License.  You may obtain a copy of the License at
+#
+#   http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied.  See the License for the
+# specific language governing permissions and limitations
+# under the License.
+
+{
+   # `testthat::skip()`s cause a valgrind error that does not show up on CRAN. 
+   <testthat_skip_error>
+   Memcheck:Cond
+   fun:gregexpr_Regexc
+   fun:do_regexpr
+   fun:bcEval
+   fun:Rf_eval
+   fun:R_execClosure
+   fun:Rf_applyClosure
+   fun:bcEval
+   fun:Rf_eval
+   fun:forcePromise
+   fun:FORCE_PROMISE
+   fun:getvar
+   fun:bcEval
+}
diff --git a/ci/scripts/r_valgrind.sh b/ci/scripts/r_valgrind.sh
new file mode 100755
index 000000000000..43f8c26739af
--- /dev/null
+++ b/ci/scripts/r_valgrind.sh
@@ -0,0 +1,45 @@
+#!/usr/bin/env bash
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements.  See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership.  The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License.  You may obtain a copy of the License at
+#
+#   http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied.  See the License for the
+# specific language governing permissions and limitations
+# under the License.
+
+set -ex
+
+: ${R_BIN:=RDvalgrind}
+
+source_dir=${1}/r
+
+${R_BIN} CMD INSTALL ${source_dir}
+pushd ${source_dir}/tests
+
+export TEST_R_WITH_ARROW=TRUE
+# to generate suppression files run:
+# ${R_BIN} --vanilla -d "valgrind --tool=memcheck --leak-check=full --track-origins=yes --gen-suppressions=all --log-file=memcheck.log" -f testtthat.supp
+${R_BIN} --vanilla -d "valgrind --tool=memcheck --leak-check=full --track-origins=yes --suppressions=/${1}/ci/etc/valgrind-cran.supp" -f testthat.R |& tee testthat.out
+
+# valgrind --error-exitcode=1 should return an erroring exit code that we can catch,
+# but R eats that and returns 0, so we need to look at the output and make sure that
+# we have 0 errors instead.
+if [ $(grep -c "ERROR SUMMARY: 0 errors" testthat.out) != 1 ]; then
+  cat testthat.out
+  echo "Found Valgrind errors"
+  exit 1
+fi
+
+# We might also considering using the greps that LibthGBM uses:
+# https://github.com/microsoft/LightGBM/blob/fa6d356555f9ef888acf5f5e259dca958ca24f6d/.ci/test_r_package_valgrind.sh#L20-L85
+
+popd
\ No newline at end of file
diff --git a/dev/tasks/tasks.yml b/dev/tasks/tasks.yml
index f82960bf3537..832d16d9f69b 100644
--- a/dev/tasks/tasks.yml
+++ b/dev/tasks/tasks.yml
@@ -862,6 +862,14 @@ tasks:
         FEDORA: 33
       run: fedora-python
 
+  test-r-linux-valgrind:
+    ci: azure
+    template: docker-tests/azure.linux.yml
+    params:
+      env:
+        UBUNTU: 18.04
+      run: ubuntu-r-valgrind
+
   test-r-linux-as-cran:
     ci: github
     template: r/github.linux.cran.yml
diff --git a/docker-compose.yml b/docker-compose.yml
index 4158ee3ff649..d9ca731dbd3f 100644
--- a/docker-compose.yml
+++ b/docker-compose.yml
@@ -116,6 +116,7 @@ x-hierarchy:
   - ubuntu-cpp-sanitizer
   - ubuntu-cpp-thread-sanitizer
   - ubuntu-r-sanitizer
+  - ubuntu-r-valgrind
   - python-sdist
   - r
   # helper services
@@ -1050,6 +1051,31 @@ services:
       /bin/bash -c "
         /arrow/ci/scripts/r_sanitize.sh /arrow"
 
+  ubuntu-r-valgrind:
+    # Only 18.04 and amd64 supported
+    # Usage:
+    #   docker-compose build ubuntu-r-valgrind
+    #   docker-compose run ubuntu-r-valgrind
+    image: ${REPO}:amd64-ubuntu-18.04-r-valgrind
+    build:
+      context: .
+      dockerfile: ci/docker/linux-r.dockerfile
+      cache_from:
+        - ${REPO}:amd64-ubuntu-18.04-r-valgrind
+      args:
+        base: wch1/r-debug:latest
+        r_bin: RDvalgrind
+    environment:
+      <<: *ccache
+      # AVX512 not supported by Valgrind (similar to ARROW-9851) some runners support AVX512 and some do not
+      # so some build might pass without this setting, but we want to ensure that we stay to AVX2 regardless of runner.
+      EXTRA_CMAKE_FLAGS: "-DARROW_RUNTIME_SIMD_LEVEL=AVX2"
+    volumes: *ubuntu-volumes
+    command: >
+      /bin/bash -c "
+        /arrow/ci/scripts/r_valgrind.sh /arrow"
+
+
   ################################# Go ########################################
 
   debian-go:
diff --git a/r/inst/build_arrow_static.sh b/r/inst/build_arrow_static.sh
index e9c82a10e400..6fba5be34bcf 100755
--- a/r/inst/build_arrow_static.sh
+++ b/r/inst/build_arrow_static.sh
@@ -53,8 +53,8 @@ ${CMAKE} -DARROW_BOOST_USE_SHARED=OFF \
     -DARROW_DATASET=${ARROW_DATASET:-ON} \
     -DARROW_DEPENDENCY_SOURCE=BUNDLED \
     -DARROW_FILESYSTEM=ON \
-    -DARROW_JEMALLOC=${ARROW_JEMALLOC:-ON} \
-    -DARROW_MIMALLOC=${ARROW_MIMALLOC:-$ARROW_DEFAULT_PARAM} \
+    -DARROW_JEMALLOC=${ARROW_JEMALLOC:-$ARROW_DEFAULT_PARAM} \
+    -DARROW_MIMALLOC=${ARROW_MIMALLOC:-ON} \
     -DARROW_JSON=ON \
     -DARROW_PARQUET=${ARROW_PARQUET:-ON} \
     -DARROW_S3=${ARROW_S3:-$ARROW_DEFAULT_PARAM} \
diff --git a/r/tests/testthat/helper-skip.R b/r/tests/testthat/helper-skip.R
index 6fb97da000d8..b1c7d66bec84 100644
--- a/r/tests/testthat/helper-skip.R
+++ b/r/tests/testthat/helper-skip.R
@@ -22,6 +22,11 @@ build_features <- c(
 )
 
 skip_if_not_available <- function(feature) {
+  if (feature == "re2") {
+    # RE2 does not support valgrind (on purpose): https://github.com/google/re2/issues/177
+    skip_on_valgrind()
+  }
+
   yes <- feature %in% names(build_features) && build_features[feature]
   if (!yes) {
     skip(paste("Arrow C++ not built with", feature))
@@ -29,6 +34,8 @@ skip_if_not_available <- function(feature) {
 }
 
 skip_if_no_pyarrow <- function() {
+  skip_on_valgrind()
+
   skip_if_not_installed("reticulate")
   if (!reticulate::py_module_available("pyarrow")) {
     skip("pyarrow not available for testing")
@@ -49,6 +56,18 @@ skip_if_not_running_large_memory_tests <- function() {
   )
 }
 
+skip_on_valgrind <- function() {
+  # This does not actually skip on valgrind because we can't exactly detect it.
+  # Instead, it skips on CRAN when the OS is linux + and the R version is development 
+  # (which is where valgrind is run as of this code)
+  linux_dev <- identical(tolower(Sys.info()[["sysname"]]), "linux") &&
+    grepl("devel", R.version.string)
+
+  if (linux_dev) {
+    skip_on_cran()
+  }
+}
+
 process_is_running <- function(x) {
   cmd <- sprintf("ps aux | grep '%s' | grep -v grep", x)
   tryCatch(system(cmd, ignore.stdout = TRUE) == 0, error = function(e) FALSE)
diff --git a/r/tests/testthat/test-Array.R b/r/tests/testthat/test-Array.R
index e064f81cdfaf..26d0a3005e42 100644
--- a/r/tests/testthat/test-Array.R
+++ b/r/tests/testthat/test-Array.R
@@ -53,6 +53,7 @@ test_that("binary Array", {
   expect_array_roundtrip(bin, fixed_size_binary(byte_width = 10))
 
   # degenerate cases
+  skip_on_valgrind() # valgrind errors on these tests ARROW-12638
   bin <- vctrs::new_vctr(
     list(1:10),
     class = "arrow_binary"
diff --git a/r/tests/testthat/test-arrow.R b/r/tests/testthat/test-arrow.R
index 2c901e1c96a0..2ab127e48000 100644
--- a/r/tests/testthat/test-arrow.R
+++ b/r/tests/testthat/test-arrow.R
@@ -62,6 +62,10 @@ test_that("check for an ArrowObject in functions use std::shared_ptr", {
 })
 
 test_that("MemoryPool calls gc() to free memory when allocation fails (ARROW-10080)", {
+  # There is a valgrind error on this test because there cannot be memory allocated
+  # which is exactly what this test is checking, but we quiet this
+  skip_on_valgrind()
+
   env <- new.env()
   trace(gc, print = FALSE, tracer = function() {
           env$gc_was_called <- TRUE
diff --git a/r/tests/testthat/test-compute-aggregate.R b/r/tests/testthat/test-compute-aggregate.R
index 0621b7779c73..51bfa05b9164 100644
--- a/r/tests/testthat/test-compute-aggregate.R
+++ b/r/tests/testthat/test-compute-aggregate.R
@@ -37,7 +37,11 @@ test_that("sum.Array", {
 
   floats <- c(floats, NA)
   na <- Array$create(floats)
-  expect_identical(as.numeric(sum(na)), sum(floats))
+  if (!grepl("devel", R.version.string)) {
+    # Valgrind on R-devel confuses NaN and NA_real_
+    # https://r.789695.n4.nabble.com/Difference-in-NA-behavior-in-R-devel-running-under-valgrind-td4768731.html
+    expect_identical(as.numeric(sum(na)), sum(floats))
+  }
   expect_r6_class(sum(na, na.rm = TRUE), "Scalar")
   expect_identical(as.numeric(sum(na, na.rm = TRUE)), sum(floats, na.rm = TRUE))
 
@@ -78,7 +82,11 @@ test_that("mean.Array", {
 
   floats <- c(floats, NA)
   na <- Array$create(floats)
-  expect_identical(as.vector(mean(na)), mean(floats))
+  if (!grepl("devel", R.version.string)) {
+    # Valgrind on R-devel confuses NaN and NA_real_
+    # https://r.789695.n4.nabble.com/Difference-in-NA-behavior-in-R-devel-running-under-valgrind-td4768731.html
+    expect_identical(as.vector(mean(na)), mean(floats))
+  }
   expect_r6_class(mean(na, na.rm = TRUE), "Scalar")
   expect_identical(as.vector(mean(na, na.rm = TRUE)), mean(floats, na.rm = TRUE))