diff --git a/src/briefcase/commands/base.py b/src/briefcase/commands/base.py
index 07d6d2c2d..880a6edad 100644
--- a/src/briefcase/commands/base.py
+++ b/src/briefcase/commands/base.py
@@ -107,6 +107,8 @@ class BaseCommand(ABC):
     cmd_line = "briefcase {command} {platform} {output_format}"
     supported_host_os = {"Darwin", "Linux", "Windows"}
     supported_host_os_reason = f"This command is not supported on {platform.system()}."
+    supported_host_arch: set()
+    supports_32bit_python = False
     # defined by platform-specific subclasses
     command: str
     description: str
@@ -484,6 +486,21 @@ def verify_host(self):
         if self.tools.host_os not in self.supported_host_os:
             raise UnsupportedHostError(self.supported_host_os_reason)
 
+        if (
+            self.supported_host_arch
+            and self.tools.host_arch not in self.supported_host_arch
+        ):
+            raise BriefcaseCommandError(
+                f"Briefcase is not supported on {platform.system()} for {platform.machine()}."
+            )
+
+        # if the machine's largest supported pointer address is less than 2^32,
+        # then assume the machine is 32 bits or Python itself is 32 bits
+        if not self.supports_32bit_python and self.tools.sys.maxsize <= 2**32:
+            raise BriefcaseCommandError(
+                f"This command is not supported for a 32bit {platform.machine()} or 32bit Python"
+            )
+
     def verify_tools(self):
         """Verify that the tools needed to run this Command exist.
 
diff --git a/src/briefcase/integrations/java.py b/src/briefcase/integrations/java.py
index 4f041349e..dc8f7e6b4 100644
--- a/src/briefcase/integrations/java.py
+++ b/src/briefcase/integrations/java.py
@@ -29,7 +29,9 @@ def OpenJDK_download_url(self):
         arch = {
             "x86_64": "x64",  # Linux\macOS x86-64
             "aarch64": "aarch64",  # Linux arm64
-            "armv6l": "arm",  # Linux arm
+            "armv6l": "arm",  # Linux armv6
+            "armv7l": "arm",  # Linux armv7
+            "armv8l": "arm",  # Linux armv8
             "arm64": "aarch64",  # macOS arm64
             "AMD64": "x64",  # Windows x86-64
         }.get(self.tools.host_arch)
diff --git a/src/briefcase/integrations/linuxdeploy.py b/src/briefcase/integrations/linuxdeploy.py
index 9d5d9f5f5..50b875d95 100644
--- a/src/briefcase/integrations/linuxdeploy.py
+++ b/src/briefcase/integrations/linuxdeploy.py
@@ -48,6 +48,11 @@ def download_url(self) -> str:
     def file_path(self) -> Path:
         """The folder on the local filesystem that contains the file_name."""
 
+    @property
+    def arch(self):
+        """Return architecture for LinuxDeploy."""
+        return {"i686": "i386"}.get(self.tools.host_arch, self.tools.host_arch)
+
     def exists(self) -> bool:
         return (self.file_path / self.file_name).is_file()
 
@@ -208,7 +213,7 @@ class LinuxDeployQtPlugin(LinuxDeployPluginBase, ManagedTool):
 
     @property
     def file_name(self) -> str:
-        return f"linuxdeploy-plugin-qt-{self.tools.host_arch}.AppImage"
+        return f"linuxdeploy-plugin-qt-{self.arch}.AppImage"
 
     @property
     def download_url(self) -> str:
@@ -319,7 +324,7 @@ def file_path(self) -> Path:
 
     @property
     def file_name(self) -> str:
-        return f"linuxdeploy-{self.tools.host_arch}.AppImage"
+        return f"linuxdeploy-{self.arch}.AppImage"
 
     @property
     def download_url(self) -> str:
diff --git a/src/briefcase/platforms/linux/appimage.py b/src/briefcase/platforms/linux/appimage.py
index f9916a5f4..afd3787bd 100644
--- a/src/briefcase/platforms/linux/appimage.py
+++ b/src/briefcase/platforms/linux/appimage.py
@@ -44,7 +44,8 @@ def project_path(self, app):
 
     def binary_name(self, app):
         safe_name = app.formal_name.replace(" ", "_")
-        return f"{safe_name}-{app.version}-{self.tools.host_arch}.AppImage"
+        arch = {"i686": "i386"}.get(self.tools.host_arch, self.tools.host_arch)
+        return f"{safe_name}-{app.version}-{arch}.AppImage"
 
     def binary_path(self, app):
         return self.bundle_path(app) / self.binary_name(app)
diff --git a/src/briefcase/platforms/linux/system.py b/src/briefcase/platforms/linux/system.py
index 829f4dabd..4d7fb09b6 100644
--- a/src/briefcase/platforms/linux/system.py
+++ b/src/briefcase/platforms/linux/system.py
@@ -34,6 +34,7 @@ class LinuxSystemPassiveMixin(LinuxMixin):
     # Docker exists. It is used by commands that are "passive" from the
     # perspective of the build system (e.g., Run).
     output_format = "system"
+    supports_32bit_python = True
     supported_host_os = {"Darwin", "Linux"}
     supported_host_os_reason = (
         "Linux system projects can only be built on Linux, or on macOS using Docker."
@@ -61,6 +62,7 @@ def linux_arch(self):
             "x86_64": "amd64",
             "aarch64": "arm64",
             "armv6l": "armhf",
+            "i686": "i386",
         }.get(self.tools.host_arch, self.tools.host_arch)
 
     def build_path(self, app):
diff --git a/src/briefcase/platforms/windows/__init__.py b/src/briefcase/platforms/windows/__init__.py
index fbd2c18f0..2b096eb70 100644
--- a/src/briefcase/platforms/windows/__init__.py
+++ b/src/briefcase/platforms/windows/__init__.py
@@ -19,6 +19,7 @@ class WindowsMixin:
     platform = "windows"
     supported_host_os = {"Windows"}
     supported_host_os_reason = "Windows applications can only be built on Windows."
+    supported_host_arch = {"AMD64"}
 
     def binary_path(self, app):
         return self.bundle_path(app) / self.packaging_root / f"{app.formal_name}.exe"