diff --git a/ml-proto/host/parser.mly b/ml-proto/host/parser.mly
index fe19130460..954cb46dd7 100644
--- a/ml-proto/host/parser.mly
+++ b/ml-proto/host/parser.mly
@@ -269,7 +269,7 @@ func :
 
 segment :
   | LPAR SEGMENT INT TEXT RPAR
-    { {Memory.addr = int_of_string $3; Memory.data = $4} @@ at() }
+    { {Memory.addr = Int64.of_string $3; Memory.data = $4} @@ at() }
 ;
 segment_list :
   | /* empty */ { [] }
@@ -278,10 +278,10 @@ segment_list :
 
 memory :
   | LPAR MEMORY INT INT segment_list RPAR
-    { {initial = int_of_string $3; max = int_of_string $4; segments = $5 }
+    { {initial = Int64.of_string $3; max = Int64.of_string $4; segments = $5}
         @@ at() }
   | LPAR MEMORY INT segment_list RPAR
-    { {initial = int_of_string $3; max = int_of_string $3; segments = $4 }
+    { {initial = Int64.of_string $3; max = Int64.of_string $3; segments = $4}
         @@ at() }
 ;
 
diff --git a/ml-proto/spec/ast.ml b/ml-proto/spec/ast.ml
index 4dc34b28bf..27f1697945 100644
--- a/ml-proto/spec/ast.ml
+++ b/ml-proto/spec/ast.ml
@@ -114,8 +114,8 @@ and arm' =
 type memory = memory' Source.phrase
 and memory' =
 {
-  initial : Memory.size;
-  max : Memory.size;
+  initial : int64;
+  max : int64;
   segments : segment list;
 }
 and segment = Memory.segment Source.phrase
diff --git a/ml-proto/spec/check.ml b/ml-proto/spec/check.ml
index 921dcb8766..28afc891f0 100644
--- a/ml-proto/spec/check.ml
+++ b/ml-proto/spec/check.ml
@@ -306,7 +306,8 @@ let check_export c set ex =
   NameSet.add name set
 
 let check_segment size prev_end seg =
-  let seg_end = seg.it.Memory.addr + String.length seg.it.Memory.data in
+  let seg_len = Int64.of_int (String.length seg.it.Memory.data) in
+  let seg_end = Int64.add seg.it.Memory.addr seg_len in
   require (seg.it.Memory.addr >= prev_end) seg.at
     "data segment not disjoint and ordered";
   require (size >= seg_end) seg.at
@@ -314,9 +315,12 @@ let check_segment size prev_end seg =
   seg_end
 
 let check_memory memory =
-  require (memory.it.initial <= memory.it.max) memory.at
+  let mem = memory.it in
+  require (mem.initial <= mem.max) memory.at
     "initial memory size must be less than maximum";
-  ignore (List.fold_left (check_segment memory.it.initial) 0 memory.it.segments)
+  require (mem.max <= 4294967296L) memory.at
+    "linear memory size must be less or equal to 4GiB";
+  ignore (List.fold_left (check_segment mem.initial) Int64.zero mem.segments)
 
 let check_module m =
   let {imports; exports; tables; funcs; memory} = m.it in
diff --git a/ml-proto/spec/eval.ml b/ml-proto/spec/eval.ml
index 61e011a112..bc84b8094c 100644
--- a/ml-proto/spec/eval.ml
+++ b/ml-proto/spec/eval.ml
@@ -14,7 +14,7 @@ let error = Error.error
 type value = Values.value
 type func = Ast.func
 type import = value list -> value option
-type host_params = {page_size : Memory.size}
+type host_params = {page_size : int}
 
 module ExportMap = Map.Make(String)
 type export_map = func ExportMap.t
@@ -42,9 +42,6 @@ type config =
   return : label
 }
 
-let page_size c =
-  I32.of_int32 (Int32.of_int c.module_.host.page_size)
-
 let lookup category list x =
   try List.nth list x.it with Failure _ ->
     error x.at ("runtime: undefined " ^ category ^ " " ^ string_of_int x.it)
@@ -103,6 +100,11 @@ let int32 v at =
   | Int32 i -> i
   | v -> type_error at v Types.Int32Type
 
+let mem_size v at =
+  let i32 = int32 v at in
+  let i64 = Int64.of_int32 i32 in
+  Int64.shift_right_logical (Int64.shift_left i64 32) 32
+
 
 (* Evaluation *)
 
@@ -175,30 +177,26 @@ let rec eval_expr (c : config) (e : expr) =
 
   | Load ({ty; align = _}, e1) ->
     let mem = some_memory c e.at in
-    let v1 = some (eval_expr c e1) e1.at in
-    let a = Memory.address_of_value v1 in
-    (try Some (Memory.load mem a ty) with exn -> memory_error e.at exn)
+    let v1 = mem_size (eval_expr c e1) e1.at in
+    (try Some (Memory.load mem v1 ty) with exn -> memory_error e.at exn)
 
   | Store ({ty = _; align = _}, e1, e2) ->
     let mem = some_memory c e.at in
-    let v1 = some (eval_expr c e1) e1.at in
+    let v1 = mem_size (eval_expr c e1) e1.at in
     let v2 = some (eval_expr c e2) e2.at in
-    let a = Memory.address_of_value v1 in
-    (try Memory.store mem a v2 with exn -> memory_error e.at exn);
+    (try Memory.store mem v1 v2 with exn -> memory_error e.at exn);
     Some v2
 
   | LoadExtend ({memop = {ty; align = _}; sz; ext}, e1) ->
     let mem = some_memory c e.at in
-    let v1 = some (eval_expr c e1) e1.at in
-    let a = Memory.address_of_value v1 in
-    (try Some (Memory.load_extend mem a sz ext ty) with exn -> memory_error e.at exn)
+    let v1 = mem_size (eval_expr c e1) e1.at in
+    (try Some (Memory.load_extend mem v1 sz ext ty) with exn -> memory_error e.at exn)
 
   | StoreWrap ({memop = {ty; align = _}; sz}, e1, e2) ->
     let mem = some_memory c e.at in
-    let v1 = some (eval_expr c e1) e1.at in
+    let v1 = mem_size (eval_expr c e1) e1.at in
     let v2 = some (eval_expr c e2) e2.at in
-    let a = Memory.address_of_value v1 in
-    (try Memory.store_wrap mem a sz v2 with exn -> memory_error e.at exn);
+    (try Memory.store_wrap mem v1 sz v2 with exn -> memory_error e.at exn);
     Some v2
 
   | Const v ->
@@ -237,19 +235,20 @@ let rec eval_expr (c : config) (e : expr) =
       | exn -> numerics_error e.at exn)
 
   | PageSize ->
-    Some (Int32 (page_size c))
+    Some (Int32 (Int32.of_int c.module_.host.page_size))
 
   | MemorySize ->
     let mem = some_memory c e.at in
-    Some (Int32 (I32.of_int32 (Int32.of_int (Memory.size mem))))
+    let i64 = Memory.size mem in
+    assert (i64 < Int64.of_int32 (Int32.max_int));
+    Some (Int32 (Int64.to_int32 i64))
 
   | ResizeMemory e ->
     let mem = some_memory c e.at in
-    let i = int32 (eval_expr c e) e.at in
-    if (I32.rem_u i (page_size c)) <> I32.zero then
+    let sz = mem_size (eval_expr c e) e.at in
+    if (Int64.rem sz (Int64.of_int c.module_.host.page_size)) <> 0L then
       error e.at "runtime: resize_memory operand not multiple of page_size";
-    (* TODO: The conversion to int could overflow. *)
-    Memory.resize mem (Int32.to_int i);
+    Memory.resize mem sz;
     None
 
 and eval_expr_option c eo =
diff --git a/ml-proto/spec/eval.mli b/ml-proto/spec/eval.mli
index 2442bfe13f..1a9ab090e8 100644
--- a/ml-proto/spec/eval.mli
+++ b/ml-proto/spec/eval.mli
@@ -5,7 +5,7 @@
 type instance
 type value = Values.value
 type import = value list -> value option
-type host_params = {page_size : Memory.size}
+type host_params = {page_size : int}
 
 val init : Ast.module_ -> import list -> host_params -> instance
 val invoke : instance -> string -> value list -> value option
diff --git a/ml-proto/spec/memory.ml b/ml-proto/spec/memory.ml
index 4263b9fa69..336a36ffd5 100644
--- a/ml-proto/spec/memory.ml
+++ b/ml-proto/spec/memory.ml
@@ -6,10 +6,7 @@ open Bigarray
 open Types
 open Values
 
-
-(* Types and view types *)
-
-type address = int
+type address = int64
 type size = address
 type mem_size = Mem8 | Mem16 | Mem32
 type extension = SX | ZX
@@ -21,15 +18,37 @@ type memory' = (int, int8_unsigned_elt, c_layout) Array1.t
 type memory = memory' ref
 type t = memory
 
-
-(* Creation and initialization *)
-
 exception Type
 exception Bounds
 exception Address
 
+(*
+ * These limitations should be considered part of the host environment and not
+ * part of the spec defined by this file.
+ * ==========================================================================
+ *)
+
+let host_size_of_int64 n =
+  if n < Int64.zero || n > (Int64.of_int max_int) then raise Out_of_memory;
+  Int64.to_int n
+
+let int64_of_host_size n =
+  Int64.of_int n
+
+let host_index_of_int64 a n =
+  assert (n >= 0);
+  let n' = Int64.of_int n in
+  if (a < Int64.zero) ||
+     (Int64.sub Int64.max_int a < n') ||
+     (Int64.add a n' > Int64.of_int max_int) then raise Bounds;
+  Int64.to_int a
+
+(* ========================================================================== *)
+
+
 let create' n =
-  let mem = Array1.create Int8_unsigned C_layout n in
+  let sz = host_size_of_int64 n in
+  let mem = Array1.create Int8_unsigned C_layout sz in
   Array1.fill mem 0;
   mem
 
@@ -38,48 +57,45 @@ let create n =
 
 let init_seg mem seg =
   (* There currently is no way to blit from a string. *)
-  for i = 0 to String.length seg.data - 1 do
-    !mem.{seg.addr + i} <- Char.code seg.data.[i]
+  let n = String.length seg.data in
+  let base = host_index_of_int64 seg.addr n in
+  for i = 0 to n - 1 do
+    !mem.{base + i} <- Char.code seg.data.[i]
   done
 
 let init mem segs =
   try List.iter (init_seg mem) segs with Invalid_argument _ -> raise Bounds
 
-
 let size mem =
-  Array1.dim !mem
+  int64_of_host_size (Array1.dim !mem)
 
 let resize mem n =
-  let before = !mem in
   let after = create' n in
-  let min = min (Array1.dim before) n in
-  Array1.blit (Array1.sub before 0 min) (Array1.sub after 0 min);
+  let min = host_index_of_int64 (min (size mem) n) 0 in
+  Array1.blit (Array1.sub !mem 0 min) (Array1.sub after 0 min);
   mem := after
 
-open Values
-
-(* TODO: The conversion to int could overflow *)
-let address_of_value = function
-  | Int32 i -> Int32.to_int (I32.to_int32 i)
-  | _ -> raise Address
-
-
-(* Load and store *)
-
 let rec loadn mem n a =
   assert (n > 0 && n <= 8);
-  let byte = try Int64.of_int !mem.{a} with Invalid_argument _ -> raise Bounds in
+  let i = host_index_of_int64 a n in
+  try loadn' mem n i with Invalid_argument _ -> raise Bounds
+
+and loadn' mem n i =
+  let byte = Int64.of_int !mem.{i} in
   if n = 1 then
     byte
   else
-    Int64.logor byte (Int64.shift_left (loadn mem (n-1) (a+1)) 8)
+    Int64.logor byte (Int64.shift_left (loadn' mem (n-1) (i+1)) 8)
 
 let rec storen mem n a v =
   assert (n > 0 && n <= 8);
-  let byte = (Int64.to_int v) land 255 in
-  (try !mem.{a} <- byte with Invalid_argument _ -> raise Bounds);
+  let i = host_index_of_int64 a n in
+  try storen' mem n i v with Invalid_argument _ -> raise Bounds
+
+and storen' mem n i v =
+  !mem.{i} <- (Int64.to_int v) land 255;
   if (n > 1) then
-    storen mem (n-1) (a+1) (Int64.shift_right v 8)
+    storen' mem (n-1) (i+1) (Int64.shift_right v 8)
 
 let load mem a t =
   match t with
diff --git a/ml-proto/spec/memory.mli b/ml-proto/spec/memory.mli
index 6f7112a29f..2b3320a00d 100644
--- a/ml-proto/spec/memory.mli
+++ b/ml-proto/spec/memory.mli
@@ -4,7 +4,7 @@
 
 type memory
 type t = memory
-type address = int
+type address = int64
 type size = address
 type mem_size = Mem8 | Mem16 | Mem32
 type extension = SX | ZX
@@ -24,5 +24,3 @@ val load : memory -> address -> value_type -> value
 val store : memory -> address -> value -> unit
 val load_extend : memory -> address -> mem_size -> extension -> value_type -> value
 val store_wrap : memory -> address -> mem_size -> value -> unit
-
-val address_of_value : Values.value -> address