diff --git a/src/cff.cc b/src/cff.cc
index af54a4a1..af6d1eb8 100644
--- a/src/cff.cc
+++ b/src/cff.cc
@@ -1250,6 +1250,11 @@ bool OpenTypeCFF::Parse(const uint8_t *data, size_t length) {
   if (!maxp) {
     return Error("Required maxp table missing");
   }
+  if (maxp->version_1) {
+    Warning("Fixing incorrect maxp version for CFF font");
+    maxp->version_1 = false;
+  }
+
   const uint16_t num_glyphs = maxp->num_glyphs;
   const size_t sid_max = string_index.count + kNStdString;
   // string_index.count == 0 is allowed.
diff --git a/src/maxp.cc b/src/maxp.cc
index 232e4a98..59b98cd7 100644
--- a/src/maxp.cc
+++ b/src/maxp.cc
@@ -29,6 +29,23 @@ bool OpenTypeMAXP::Parse(const uint8_t *data, size_t length) {
     return Error("numGlyphs is 0");
   }
 
+  // Per https://learn.microsoft.com/en-gb/typography/opentype/spec/maxp,
+  // the only two 'maxp' version numbers are 0.5 (for CFF/CFF2) and 1.0
+  // (for TrueType).
+  if (version == 0x00005000) {
+    this->version_1 = false;
+    return true;
+  }
+
+  if (version != 0x00010000) {
+    Warning("Unrecognized version %08x, attempting to fix", version);
+    // If there's at least enough data to read as a version 1.0 table,
+    // we'll try that.
+    if (length >= 32) {
+      version = 0x00010000;
+    }
+  }
+
   if (version >> 16 == 1) {
     this->version_1 = true;
     if (!table.ReadU16(&this->max_points) ||
@@ -47,19 +64,15 @@ bool OpenTypeMAXP::Parse(const uint8_t *data, size_t length) {
       return Error("Failed to read version 1 table data");
     }
 
-    if (this->max_zones == 0) {
+    if (this->max_zones < 1) {
       // workaround for ipa*.ttf Japanese fonts.
       Warning("Bad maxZones: %u", this->max_zones);
       this->max_zones = 1;
-    } else if (this->max_zones == 3) {
-      // workaround for Ecolier-*.ttf fonts.
+    } else if (this->max_zones > 2) {
+      // workaround for Ecolier-*.ttf fonts and bad fonts in some PDFs
       Warning("Bad maxZones: %u", this->max_zones);
       this->max_zones = 2;
     }
-
-    if ((this->max_zones != 1) && (this->max_zones != 2)) {
-      return Error("Bad maxZones: %u", this->max_zones);
-    }
   } else {
     this->version_1 = false;
   }