diff --git a/spacy/tests/pipeline/test_span_predictor.py b/spacy/tests/pipeline/test_span_predictor.py
new file mode 100644
index 000000000..1adaecd3f
--- /dev/null
+++ b/spacy/tests/pipeline/test_span_predictor.py
@@ -0,0 +1,129 @@
+import pytest
+import spacy
+
+from spacy import util
+from spacy.training import Example
+from spacy.lang.en import English
+from spacy.tests.util import make_tempdir
+from spacy.ml.models.coref_util import (
+    DEFAULT_CLUSTER_PREFIX,
+    select_non_crossing_spans,
+    get_sentence_ids,
+    spans2ints,
+)
+
+from thinc.util import has_torch
+
+# fmt: off
+TRAIN_DATA = [
+    (
+        "John Smith picked up the red ball and he threw it away.",
+        {
+            "spans": {
+                f"{DEFAULT_CLUSTER_PREFIX}_1": [
+                    (0, 11, "MENTION"),      # John Smith
+                    (38, 41, "MENTION"),     # he
+
+                ],
+                f"{DEFAULT_CLUSTER_PREFIX}_2": [
+                    (25, 33, "MENTION"),     # red ball
+                    (47, 50, "MENTION"),     # it
+                ],
+                f"coref_head_clusters_1": [
+                    (5, 11, "MENTION"),      # Smith
+                    (38, 41, "MENTION"),     # he
+
+                ],
+                f"coref_head_clusters_2": [
+                    (29, 33, "MENTION"),     # red ball
+                    (47, 50, "MENTION"),     # it
+                ]
+            }
+        },
+    ),
+]
+# fmt: on
+
+
+@pytest.fixture
+def nlp():
+    return English()
+
+
+@pytest.fixture
+def snlp():
+    en = English()
+    en.add_pipe("sentencizer")
+    return en
+
+
+@pytest.mark.skipif(not has_torch, reason="Torch not available")
+def test_add_pipe(nlp):
+    nlp.add_pipe("span_predictor")
+    assert nlp.pipe_names == ["span_predictor"]
+
+
+@pytest.mark.skipif(not has_torch, reason="Torch not available")
+def test_not_initialized(nlp):
+    nlp.add_pipe("span_predictor")
+    text = "She gave me her pen."
+    with pytest.raises(ValueError, match="E109"):
+        nlp(text)
+
+
+@pytest.mark.skipif(not has_torch, reason="Torch not available")
+def test_span_predictor_serialization(nlp):
+    # Test that the span predictor component can be serialized
+    nlp.add_pipe("span_predictor", last=True)
+    nlp.initialize()
+    assert nlp.pipe_names == ["span_predictor"]
+    text = "She gave me her pen."
+    doc = nlp(text)
+
+    with make_tempdir() as tmp_dir:
+        nlp.to_disk(tmp_dir)
+        nlp2 = spacy.load(tmp_dir)
+        assert nlp2.pipe_names == ["span_predictor"]
+        doc2 = nlp2(text)
+
+        assert spans2ints(doc) == spans2ints(doc2)
+
+
+@pytest.mark.skipif(not has_torch, reason="Torch not available")
+def test_overfitting_IO(nlp):
+    # Simple test to try and quickly overfit - ensuring the ML models work correctly
+    train_examples = []
+    for text, annot in TRAIN_DATA:
+        train_examples.append(Example.from_dict(nlp.make_doc(text), annot))
+
+    nlp.add_pipe("span_predictor")
+    optimizer = nlp.initialize()
+    test_text = TRAIN_DATA[0][0]
+    doc = nlp(test_text)
+
+    # Needs ~12 epochs to converge
+    for i in range(15):
+        losses = {}
+        nlp.update(train_examples, sgd=optimizer, losses=losses)
+        doc = nlp(test_text)
+
+    # test the trained model
+    doc = nlp(test_text)
+
+    # Also test the results are still the same after IO
+    with make_tempdir() as tmp_dir:
+        nlp.to_disk(tmp_dir)
+        nlp2 = util.load_model_from_path(tmp_dir)
+        doc2 = nlp2(test_text)
+
+    # Make sure that running pipe twice, or comparing to call, always amounts to the same predictions
+    texts = [
+        test_text,
+        "I noticed many friends around me",
+        "They received it. They received the SMS.",
+    ]
+    docs1 = list(nlp.pipe(texts))
+    docs2 = list(nlp.pipe(texts))
+    docs3 = [nlp(text) for text in texts]
+    assert spans2ints(docs1[0]) == spans2ints(docs2[0])
+    assert spans2ints(docs1[0]) == spans2ints(docs3[0])