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;
   }
diff --git a/src/ots.cc b/src/ots.cc
index cefd390a..ed22b8cc 100644
--- a/src/ots.cc
+++ b/src/ots.cc
@@ -762,12 +762,18 @@ bool ProcessGeneric(ots::FontFile *header,
   ots::Table *loca = font->GetTable(OTS_TAG_LOCA);
   ots::Table *cff  = font->GetTable(OTS_TAG_CFF);
   ots::Table *cff2 = font->GetTable(OTS_TAG_CFF2);
+  ots::OpenTypeMAXP *maxp = static_cast<ots::OpenTypeMAXP*>(
+    font->GetTypedTable(OTS_TAG_MAXP));
 
   if (glyf && loca) {
     if (font->version != 0x000010000) {
       OTS_WARNING_MSG_HDR("wrong sfntVersion for glyph data");
       font->version = 0x000010000;
     }
+    if (!maxp->version_1) {
+      return OTS_FAILURE_MSG_TAG("wrong maxp version for glyph data",
+                                 OTS_TAG_MAXP);
+    }
     if (cff)
        cff->Drop("font contains both CFF and glyf/loca tables");
     if (cff2)
@@ -781,6 +787,10 @@ bool ProcessGeneric(ots::FontFile *header,
        glyf->Drop("font contains both CFF and glyf tables");
     if (loca)
        loca->Drop("font contains both CFF and loca tables");
+    if (maxp->version_1) {
+      OTS_WARNING_MSG_HDR("fixing incorrect maxp version for CFF font");
+      maxp->version_1 = false;
+    }
   } else if (font->GetTable(OTS_TAG('C','B','D','T')) &&
              font->GetTable(OTS_TAG('C','B','L','C'))) {
       // We don't sanitize bitmap tables, but don’t reject bitmap-only fonts if