diff --git a/std/crypto/chacha20.zig b/std/crypto/chacha20.zig
index 78ef700bdf..796b5c6e4f 100644
--- a/std/crypto/chacha20.zig
+++ b/std/crypto/chacha20.zig
@@ -142,7 +142,7 @@ pub fn chaCha20With64BitNonce(out: []u8, in: []const u8, counter: u64, key: [32]
     assert(in.len >= out.len);
     assert(counter +% (in.len >> 6) >= counter);
 
-    var cursor: u64 = 0;
+    var cursor: usize = 0;
     var k: [8]u32 = undefined;
     var c: [4]u32 = undefined;
 
@@ -161,7 +161,8 @@ pub fn chaCha20With64BitNonce(out: []u8, in: []const u8, counter: u64, key: [32]
     c[3] = mem.readIntSliceLittle(u32, nonce[4..8]);
 
     const block_size = (1 << 6);
-    const big_block = (block_size << 32);
+    // The full block size is greater than the address space on a 32bit machine
+    const big_block = if (@sizeOf(usize) > 4) (block_size << 32) else maxInt(usize);
 
     // first partial big block
     if (((@intCast(u64, maxInt(u32) - @truncate(u32, counter)) + 1) << 6) < in.len) {
diff --git a/std/heap.zig b/std/heap.zig
index 0751e26e7b..4a7a33772f 100644
--- a/std/heap.zig
+++ b/std/heap.zig
@@ -603,7 +603,7 @@ test "ArenaAllocator" {
     try testAllocatorAlignedShrink(&arena_allocator.allocator);
 }
 
-var test_fixed_buffer_allocator_memory: [40000 * @sizeOf(usize)]u8 = undefined;
+var test_fixed_buffer_allocator_memory: [40000 * @sizeOf(u64)]u8 = undefined;
 test "FixedBufferAllocator" {
     var fixed_buffer_allocator = FixedBufferAllocator.init(test_fixed_buffer_allocator_memory[0..]);
 
diff --git a/std/os/linux.zig b/std/os/linux.zig
index 425df3a55d..8471a1861b 100644
--- a/std/os/linux.zig
+++ b/std/os/linux.zig
@@ -1101,8 +1101,8 @@ pub fn sigaction(sig: u6, noalias act: *const Sigaction, noalias oact: ?*Sigacti
 
 const NSIG = 65;
 const sigset_t = [128 / @sizeOf(usize)]usize;
-const all_mask = []usize{maxInt(usize)};
-const app_mask = []usize{0xfffffffc7fffffff};
+const all_mask = []u32{0xffffffff, 0xffffffff};
+const app_mask = []u32{0xfffffffc, 0x7fffffff};
 
 const k_sigaction = extern struct {
     handler: extern fn (i32) void,
diff --git a/std/os/time.zig b/std/os/time.zig
index d742c5d03b..abb6412843 100644
--- a/std/os/time.zig
+++ b/std/os/time.zig
@@ -3,6 +3,7 @@ const builtin = @import("builtin");
 const Os = builtin.Os;
 const debug = std.debug;
 const testing = std.testing;
+const math = std.math;
 
 const windows = std.os.windows;
 const linux = std.os.linux;
@@ -12,33 +13,34 @@ const posix = std.os.posix;
 
 pub const epoch = @import("epoch.zig");
 
-/// Sleep for the specified duration
+/// Spurious wakeups are possible and no precision of timing is guaranteed.
 pub fn sleep(nanoseconds: u64) void {
     switch (builtin.os) {
         Os.linux, Os.macosx, Os.ios, Os.freebsd, Os.netbsd => {
             const s = nanoseconds / ns_per_s;
             const ns = nanoseconds % ns_per_s;
-            posixSleep(@intCast(u63, s), @intCast(u63, ns));
+            posixSleep(s, ns);
         },
         Os.windows => {
             const ns_per_ms = ns_per_s / ms_per_s;
             const milliseconds = nanoseconds / ns_per_ms;
-            windows.Sleep(@intCast(windows.DWORD, milliseconds));
+            const ms_that_will_fit = std.math.cast(windows.DWORD, milliseconds) catch std.math.maxInt(windows.DWORD);
+            windows.Sleep(ms_that_will_fit);
         },
         else => @compileError("Unsupported OS"),
     }
 }
 
-pub fn posixSleep(seconds: u63, nanoseconds: u63) void {
+/// Spurious wakeups are possible and no precision of timing is guaranteed.
+pub fn posixSleep(seconds: u64, nanoseconds: u64) void {
     var req = posix.timespec{
-        .tv_sec = seconds,
-        .tv_nsec = nanoseconds,
+        .tv_sec = std.math.cast(isize, seconds) catch std.math.maxInt(isize),
+        .tv_nsec = std.math.cast(isize, nanoseconds) catch std.math.maxInt(isize),
     };
     var rem: posix.timespec = undefined;
     while (true) {
         const ret_val = posix.nanosleep(&req, &rem);
         const err = posix.getErrno(ret_val);
-        if (err == 0) return;
         switch (err) {
             posix.EFAULT => unreachable,
             posix.EINVAL => {
@@ -50,6 +52,7 @@ pub fn posixSleep(seconds: u63, nanoseconds: u63) void {
                 req = rem;
                 continue;
             },
+            // This prong handles success as well as unexpected errors.
             else => return,
         }
     }
diff --git a/std/rand.zig b/std/rand.zig
index a2fdfed6fd..4a6563f65a 100644
--- a/std/rand.zig
+++ b/std/rand.zig
@@ -768,10 +768,10 @@ pub const Isaac64 = struct {
         const x = self.m[base + m1];
         self.a = mix +% self.m[base + m2];
 
-        const y = self.a +% self.b +% self.m[(x >> 3) % self.m.len];
+        const y = self.a +% self.b +% self.m[@intCast(usize, (x >> 3) % self.m.len)];
         self.m[base + m1] = y;
 
-        self.b = x +% self.m[(y >> 11) % self.m.len];
+        self.b = x +% self.m[@intCast(usize, (y >> 11) % self.m.len)];
         self.r[self.r.len - 1 - base - m1] = self.b;
     }