diff --git a/src/io/csv.js b/src/io/csv.js
index 9dda273e8b..e573dd8fee 100644
--- a/src/io/csv.js
+++ b/src/io/csv.js
@@ -133,8 +133,9 @@ export function parse(csv, options, reviver = v => v) {
     }
   }
 
-  // flush the last value
-  if (ctx.entry.length !== 0) {
+  // flush the last value. In a one-column row, the pending value is still in
+  // ctx.value with ctx.entry empty, so checking the entry alone drops it.
+  if (state !== 0 || ctx.entry.length !== 0) {
     valueEnd(ctx);
     entryEnd(ctx);
   }
diff --git a/src/io/files.js b/src/io/files.js
index 612f44b905..3c4490384b 100644
--- a/src/io/files.js
+++ b/src/io/files.js
@@ -511,9 +511,10 @@ function files(p5, fn) {
         });
 
         if (header) {
-          ret.columns = data.shift();
+          ret.columns = data.shift() || [];
         } else {
-          ret.columns = Array(data[0].length).fill(null);
+          // An empty file parses to no rows at all.
+          ret.columns = Array(data.length ? data[0].length : 0).fill(null);
         }
 
         data.forEach(line => {
diff --git a/test/unit/assets/csv_empty.csv b/test/unit/assets/csv_empty.csv
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/test/unit/assets/csv_one_column.csv b/test/unit/assets/csv_one_column.csv
new file mode 100644
index 0000000000..afd0bc93b1
--- /dev/null
+++ b/test/unit/assets/csv_one_column.csv
@@ -0,0 +1,3 @@
+alice
+bob
+carol
\ No newline at end of file
diff --git a/test/unit/assets/csv_one_value.csv b/test/unit/assets/csv_one_value.csv
new file mode 100644
index 0000000000..f70d7bba4a
--- /dev/null
+++ b/test/unit/assets/csv_one_value.csv
@@ -0,0 +1 @@
+42
\ No newline at end of file
diff --git a/test/unit/io/csv.js b/test/unit/io/csv.js
new file mode 100644
index 0000000000..a708066933
--- /dev/null
+++ b/test/unit/io/csv.js
@@ -0,0 +1,34 @@
+import { parse } from '../../../src/io/csv';
+
+suite('csv parse', function () {
+  test('keeps the last row of a one-column file without a trailing newline', function () {
+    expect(parse('a\nb')).toEqual([['a'], ['b']]);
+  });
+
+  test('parses a single value as one row', function () {
+    expect(parse('only')).toEqual([['only']]);
+    expect(parse('42')).toEqual([['42']]);
+  });
+
+  test('parses a single quoted value as one row', function () {
+    expect(parse('"only"')).toEqual([['only']]);
+  });
+
+  test('keeps a row after an empty line', function () {
+    expect(parse('a\n\nb')).toEqual([['a'], [''], ['b']]);
+  });
+
+  test('does not add a row for a trailing newline', function () {
+    expect(parse('a\n')).toEqual([['a']]);
+    expect(parse('x,y\n1,2\n')).toEqual([['x', 'y'], ['1', '2']]);
+  });
+
+  test('keeps multi-column rows and trailing empty fields', function () {
+    expect(parse('x,y\n1,2')).toEqual([['x', 'y'], ['1', '2']]);
+    expect(parse('x,')).toEqual([['x', '']]);
+  });
+
+  test('returns no rows for an empty string', function () {
+    expect(parse('')).toEqual([]);
+  });
+});
diff --git a/test/unit/io/loadTable.js b/test/unit/io/loadTable.js
index 819e4d4f16..0cddb58877 100644
--- a/test/unit/io/loadTable.js
+++ b/test/unit/io/loadTable.js
@@ -72,6 +72,23 @@ suite('loadTable', function () {
     assert.strictEqual(table.getRow(1).getNum(1), 31);
   });
 
+  test('keeps the last row of a one-column file without a trailing newline', async () => {
+    const table = await mockP5Prototype.loadTable('/test/unit/assets/csv_one_column.csv');
+    assert.equal(table.getRowCount(), 3);
+    assert.strictEqual(table.getRow(2).getString(0), 'carol');
+  });
+
+  test('loads a file holding a single value', async () => {
+    const table = await mockP5Prototype.loadTable('/test/unit/assets/csv_one_value.csv');
+    assert.equal(table.getRowCount(), 1);
+    assert.strictEqual(table.getRow(0).getNum(0), 42);
+  });
+
+  test('loads an empty file as an empty table', async () => {
+    const table = await mockP5Prototype.loadTable('/test/unit/assets/csv_empty.csv');
+    assert.equal(table.getRowCount(), 0);
+  });
+
   test('using the header option works', async () => {
     const table = await mockP5Prototype.loadTable(validFile, ',', true);
     assert.equal(table.getRowCount(), 3);